Daemons

Yesterday, we went to see the movie 'The Golden Compass' based on the book by Philip Pullman. I had read the book quite some while ago, long enough to have forgotten most of it except for the fact that I didn't like it, and therefore never bothered to get the two other parts of the trilogy. Upon seeing the movie I recalled why I disliked the book, so if you were looking for praise of your new fav movie you're on the wrong blog. The movie adaption is as far as I can tell excellent, but I generally don't like a) movies with animals, b) movies with children, and c) movies with an overdose of moral and messages imposed on the innocent audience, so was destined to dislike The Golden Compass. If you translate movie's animals and children into underdeveloped characters, this disliking of mine applies to books as well. Combine that with the fact that Pullman's writing style didn't exactly strike me as outstanding, I keep wondering what people find so great about it.

It's not like I think the story is generally bad, it is without doubt a burst of creativity and imagination. I just think he mixed up too many things on the expenses of authentic persons and consistent storytelling. Take Lyra, the hero of the story, who is supposed to save not only her friend who has been kidnapped, but the whole world, the universe, and while we're at it, also all other parallel universes. Wouldn't one think she spends some time trying to cope with that news? But no, she just goes to the North pole, while other more or less flat characters drop in and out of the storyline. That's even worse than Frodo in the Lord of the Rings, which I disliked for essentially the same reason, but at least he wonders about his destiny - Rowling's Potter does so to a much more appropriate extend, and Harry develops some personality.

Either way, the idea that people's souls live outside their bodies in animal 'daemons' is interesting. Though here too I find the setting somehow insufficiently explored. I mean, are there other animals in that world (except ice bears!), and how do they get along with the daemons? Do the daemons get born with the humans? Can they have sex with other people's daemon's? Well, maybe some of that would be answered if I'd read the other two books.

I went home thinking that would make for a great online test, and sure enough the movie website allows you to find your own daemon, so have fun, you see mine above. I admit on taking the test twice, the first result happened to have one of my ex-boyfriend's names, and how likely is that? Gee, imagine your ex-boyfriend tied to your side for the rest of your life. It remained a tiger though, concluding "You are modest, a leader, assertive, solitary and inquisitive". As with other personality tests, it's not much of a match: I'm too modest to claim I'm modest, suspicious about people who follow me, and I am too solitary to be inquisitive. Or maybe too German, i.e. 'How are you' is about the maximum of inquisition I impose on strangers. Had I picked my daemon it would have been a dirty black cat I think.

So, in the hope to better get to know our commenters, what is your daemon?

I have to say though the movie's animations are really great. How difficult would it have been only a decade ago to get your armored bear running accross the ice with a girl on his back, and how smooth and realistic does this look nowadays. This really impresses me. I keep thinking the main reason why there seems to be an increasing amount of fantasy movies is simply that it has become possible.

Trend of arXiv submissions: Update

A couple of days ago I was wondering how the trend of average monthly submissions on the arXiv would continue this year. Here is the updated plot with the statistics from 2007:



As one sees the trends do more or less continue. The average number of submissions on the hep arXiv are shown in blue and include hep-th/hep-ph/hep-lat and hep-ex. Red is astro-ph, green is cond mat, violet is math with the pink addition being math-ph. The clear extensions are cross-links. The number of hep submission slightly increased again, so it seems the temporary drop was a fluctuation. It seems to stagnate around 730 papers/month.

I am still wondering how it comes though. Several people have argued this is to be expected and just suggests 100% participation from that community (this is also the explanation you find on the arXiv website). I don't think though this can be the full story. I would expect the average number of submissions to be roughly a product of

- the total number of people in the community
- their average productivity
- the fraction of them using the arxiv

First, it is plausible to me the fraction of people using the arxiv in the hep community is saturated in North America and Europe, but given that there are still new countries coming into the game and the arXiv is as global as can be I am not sure whether this actually holds. That is to say, globally seen I'd still expect that fraction to be increasing.

Second, since the world population generally is increasing, one should think this general trend underlies the statistics, unless the fraction of people in the hep-community relative to the total population decreases.

Think China, Japan, India. Then have a look at the submitter's affiliations and tell me one should not expect further growth. I mean, yeah, there are a lot of Germans, but we don't outweigh the rest of the world.

Taken together this means, if people stick to the arxiv either the number of people in the community doesn't grow as I'd expect, or they publish less. Another possibility, as Dylan mentioned, would be that people just submit papers to other arXivs. E.g. it might be that some stuff that used to end up on hep, now goes into physics general, or history of physics. Or, maybe now that there are more online journals that publish more or less immediately, people don't submit all their papers to the pre-print server? Anybody noticed something like this?

The other thing that I found surprising is the sheer number of math publications. Not the trend - I understand that probably that community is still not completely arxiv-ed - but gee, look at all those math papers! I always had the impression mathematicians publish only sparingly, seems I was somewhat mistaken there.

For more statistics see here.

Did you know... (VI)

... that the Eskimos have 98 words for snow?

Yeah, me too, but it's actually bullshit. One of the more useful side effects of the internet is the busting of urban legends. Though it's useful only if one actually looks for it: Googling 'Eskimo Words for Snow' gives you easily several references that explain not only where the myth comes from, but also what's wrong about it.

The brief explanation is that besides there being several 'Eskimo languages' these are polysynthetic, meaning one can put several nouns with describing adjectives together into one word -- which gives a new word. I.e. there is snow, there is frozen-snow, frozen-and-dirty-snow, frozen-and-dirty-snow-with-a-crust-that-breaks-if-one-steps-on-it, and then there is snow-on-my-outside-chair-waiting-for-springtime.

Reference: Laura Martin, American Anthropologist, Vol. 88, No. 2 (Jun., 1986), pp. 418-423

"Eskimo words are the product of extremely synthetic morphology in which all word building is accomplished by multiple suffixation [...] Furthermore, precisely identical "whole" words are unlikely to recur because the particular combination of suffixes used with a "snow" root, or any other, varies by speaker and situation as well as by syntactic role."

The paper is actually quite entertaining in the way she clarifies earlier claims ("A minimal knowledge of Eskimo grammar would have confirmed the relevance of these facts to the central hypotheses [...]" Ouch.)

Either way, I was shocked to see that the above publication is from '86, since I must have read about it repeatedly, and definitely after '86.

The interesting question is much longer will that story to survive? So, take the poll below and answer the question whether you had heard of the story that the Eskimo's have so-and-so-many words for the one English word 'snow' (the precise number of words doesn't matter)

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See also: Did you know...

• ... that the can opener was invented 48 years after invention of the can?
• ... the opposite of eloquent?
• ... why the toast is called toast?
• ... what the Baconian method is?
• ... why Google is called Google?

Trend of ArXiv Submissions

The plot above shows the average monthly submission rates on the arXiv from '91 until December last year. The colors indicate the different areas blue: hep, green- cond-mat, red: astro-ph, purple: math(+pink: math-ph). The clear extensions of the bars indicate cross-links. Note how hep stagnates and drops, while math is taking the lead. Below is the statistics for hep only



I'd be interested to see the statistics from 2007, do you think this trend continued?

You find more of the arxiv statistics here.

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This post is a latecomer to our A Plottl A Day series.

Winter Wonderloo

Photos taken in Waterloo, Ontario. (Hey, Google maps has updated the satellite photos!).

More photos.

Merry Christmas!

So, here is the promised quiz to conclude our seasonal program. It turned out to be not so easy to come up with questions that aren't immediately googleable, but still seemed answerable with an appropriate amount of effort.

1. The light deflection from the sun computed within Newtonian gravity differs from the result obtained in General Relativity by a factor ...
   
   (If your answer is smaller than one, take the inverse).



2. The ratio of these both scientist's name is the commonly used symbol for ....
   
   
   
   

   /

   
   
   [My husband complains this might be too tough, so here is a hint. The first guy is a Dutchman and you're supposed to know his name if you've made it through the physics undergrad courses. The second guy had something to do with geese. Don't bother checking the source code for the filenames.]
   



3. "I didn't attend the funeral, but I sent a nice letter saying I approved of it." ~ Mark Twain
   
   Is an example for .......
   
   Take the first four letters.



4. Two divided by zero. Always on my mind. I wouldn't normally do this kind of thing. It's a ...



5. Electromagnetism is the gauge theory of the group .... Drop the first letter.

Write down your answers in one line, add two spaces in the right places and translate into Greek.

The first correct answer wins a PI mug, see photo. And no, I didn't steal it in the kitchen, I actually bought it. (If you are not willing to provide a mailing address, or aren't interested anyhow, please do not post your answer in the comments). A Merry Christmas and nice holidays to all of you!

Update: The mug is gone, congratulations to Samuel! If you want to continue guessing, spoiler warning: the correct answer is in the comment section.

iPostDoc Playlist - Christmas Edition

I'm temporarily indulging in a certain amount of self-pity while my family is opening the sparkling wine, and I am sitting in my apartment packing my household back into boxes. For those of you who haven't followed my moving fun, due to an unfortunate time-ordering-problem of the previous tenants paired with my own stupidity, I will have to move some floors upwards during the next days. So I won't be home for Christmas this year. Otoh, I don't have to repeat my annual summary a dozen times to update all the neighbors, friends, and relatives. And no brothers will have picked away all the nougat pralines while I am blogging.

-- Richard Marx: Right Here waiting -- Oceans apart day after day, And I slowly go insane -- Michael Buble: Home -- And I know just why you could not come along with me, 'Cause this was not your dream, but you always believed in me -- Chicago: Hard to say I'm sorry -- After all that we've been through, I will make it up to you. I promise to. -- Bangles: Hazy Shade of Winter -- Time, time, time, See what's become of me, While I looked around, For my possibilities -- Thornley: So far so good -- So far so good 'cause no one knows I'm faking, I wish I could show you the toll it's taking, Sometimes I live as if there's no tomorrow, So far so good -- NIN: The Wretched -- Just a little reminder, Of all the what abouts, And all the might have, Could have beens, Another day, Some other way -- Ron Sexsmith: Imaginary Friends -- For they've gone where the action is, And they've crossed you off their list -- Coldplay: Fix you -- When you try your best, but you don't succeed, When you get what you want but not what you need, When you feel so tired but you can't sleep, Stuck in reverse? -- Enya: Only Time -- Who can say where the road goes, Where the day flows, only time? -- ABBA: Waterloo (don't miss this video, Sweden '74!) -- Waterloo. Couldn’t escape if I wanted to. --

The Unitary Triangle

[The Unitary Triangle, by the CKMfitter Group]

My husband and I, we both agreed the Unitary Triangle above is the prettiest plot of contemporary physics. Unfortunately, it turns out neither of us knows very much about the actual experiments that constrain the parameter space, so we'll have to be a bit brief in this regard and just refer you elsewhere for the details.

The quantities constrained in this plot are parameters of the Cabibbo-Kobayashi-Maskawa matrix, V_CKM, which maps the weak eigenstates of the quarks to the mass eigenstates. This matrix it not diagonal, and mixes quark flavors in a similar way how the neutrino mixing works.


Here, the primes indicate the weak eigenstates. For example, it is the d' quark that transforms into an up quark in the weak decay mediated by the charged W boson - this is, on the quark level, the process which is going on in the radioactive decay of the neutron into a proton, an electron, and an antineutrino. The CKM matrix is unitary, which means the elements have to fulfil the relation

and other similar relations, but this is the one frequently used. This is a sum of three complex numbers that add up to zero. One can draw these numbers as vectors in the complex plane, and since they sum up to zero they will have to form a closed triangle. The relative angles between two sides are the arguments of the ratios of these complex numbers

Since it's a triangle one has α+β+γ = π. In addition, one can normalize one of the sides to length one and turn it such that it lies on the real axis. The resulting triangle is what you see in the plot above, the task is now to constrain it's shape, i.e. the angles or side lengths respectively. The entries of the CKM matrix affect a lot of particle physics processes, so one has to figure out smart ways how to extract them from the data. Most experiments look in detail at the weak decay of hadrons such as Kaons and B mesons. In these decays, strange and bottom quarks decay into up and charm quarks, respectively. Moreover, some of the experiments involve penguins.

Some experiments constrain an angle with a certain precision - this you see in the plot as the light blue shaded outgoing beams. Different shades indicate confidence levels. The angle β is studied through the interference between decays of neutral B⁰ mesons; γ is more challenging to measure and requires studies of rare decay processes; and also α is difficult to measure because quantum effects interfere with direct measurements. Other experiments constrain the length of a side which results in the green and orange circles. The triangle’s left side is measured through the decay rate of bottom quarks into up and charm quark; the triangle’s right side is constrained by the rate the B⁰ meson spontaneously turns into its antiparticle. And then there is the boomerang shaped green region which results from measurements of CP violation in Kaon decays.

Taken together this data constrains the top of the diagram to lie in the region marked with the red boundary.

The CKM matrix is one of the less beautiful parts of the standard model of particle physics, despite the aesthetic appeal of the unitary triangle. There are no explanations for the entries of the matrix, which just have to be taken as some parameters of the standard model. Nevertheless, the CKM matrix is relevant to quantify CP violation, which helps to understand the predominance of matter over antimatter in the universe.

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This post is part of our 2007 advent calendar A Plottl A Day.

The Quantum Hall Effect

One of the basic rules of electricity says that in order to maintain an electrical current I in a wire, a voltage U is necessary that is proportional to the current. This is Ohm's law, U = RI, where the constant of proportionality is the resistance, R. If the wire with the current is exposed to a magnetic field, the Lorentz force acts on the wire, normal to both the direction of the current and the magnetic field. If the current flows through a sheet, as a result of the Lorentz force on the electrons in the current, a voltage can be measured that is oriented along the direction of the Lorentz force. This is the Hall voltage, named after the American physicist Edwin Herbert Hall, who was the first to observe this phenomenon. The Hall effect is used today, for example, to measure magnetic fields.

Ohm's law and the Hall effect can be understood in terms of classical physics: electrons travelling through the wire bump into the atoms of the material of the wire, which causes the resistance. However, as we know for example from the study of the electronic band structure of materials, electric current is carried by particles subject to the rules of quantum mechanics. And if the temperature of the conducting material is low enough, thus reducing the strength of thermal effects, quantum effects can become apparent, leading to such striking phenomena as superconductivity. In a superconductor, charges set in motion do not scatter, since quantum mechanics doesn't allow it, and hence, there is no resistance.

Another quite strange quantum phenomenon can be observed if the current is constrained to a very thin sheet, so that it's essentially two-dimensional, and if a strong magnetic field is applied normal to the sheet. In this case, there is a special class of states for the electrons called Landau levels, and depending of the occupancy of the Landau levels, a new macroscopic phenomenon similar to superconductivity can set in: the Quantum Hall Effect.

[Source: K. v. Klitzing, G. Dorda, M. Pepper: New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45 (1980) 494-497, Figure 1.]

The experimental data shown in the plot stem from the experiment in which the Quantum Hall effect was discovered. In the experiment, shown schematically in the inset, a thin sheet structure of semiconducting material, carrying a constant current of 1 µA, was exposed to a strong magnetic field at the temperature of liquid helium, and the voltage drop along the current and the Hall voltage normal to the current and the magnetic field have been measured. These voltages are plotted on the vertical axis, and the curves are labelled as U_H for the Hall voltage, and U_pp. The horizontal axis shows another voltage, the so-called gate voltage V_g, which squeezes the electrons in a thin, two-dimensional sheet, and thus determines the occupancy of the Landau levels. Now, with varying gate voltage, a curious phenomenon can be observed in the Hall and standard voltage: the standard voltage drops to zero several times, meaning that the current flows without resistance, as in a superconductor, and at the same time, the Hall voltage takes on constant values. There is series of step-like plateaux in the Hall voltage, where it stays constant.

Now, what looks like quite an esoteric effect - take some contrived semiconductor structure, cool it down to a few Kelvin, put it in the strongest magnetic field your high-field lab can provide - has an extremely interesting twist to it:

The Hall resistance, the quotient of Hall voltage and the current, in the step-like structures comes in a regular series, R = R_K/n, where n is an integer, and R_K, the von Klitzing constant, is R_K = 25.812807557 kΩ. And the really fascinating thing is that as a consequence of the theory of Landau levels, the von Klitzing constant does not depend on the details of the material, but is universal: It is given by R = h/e² = μ₀ c/2α, where h is Planck's constant, e is the charge of the electron, and α = 1/137 is the fine structure constant, the coupling constant of electrodynamics.

Thus, the quantum Hall effect can be used, for example, to set a standard for electrical resistance, or as a means to measure the fine structure constant. Klaus von Klitzing, the German physicist who discovered the effect in February 1980 and immediately grasped these implications, was awarded the Nobel Prize in Physics 1985 for the discovery of the quantized Hall effect.

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Klaus von Klitzing's Nobel Lecture The Quantized Hall Effect gives a good introduction into theory and experiment of the Quantum Hall Effect (PDF file).

More details can be found in the proceedings of the Poincaré Seminar of November 13, 2004, dedicated to the Quantum Hall Effect. Among the talks:

Klaus von Klitzing: 25 Years of Quantum Hall Effect (QHE) A Personal View on the Discovery, Physics and Applications of this Quantum Effect (PDF file)

Benoît Douçot and Vincent Pasquier: Physics in a Strong Magnetic Field (PDF file), about Landau levels etc...

Beat Jeckelmann and Blaise Jeanneret: The Quantum Hall Effect as an Electrical Resistance Standard (PDF file), about the implications of the Quantum Hall Effect for metrology.

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This post is part of our 2007 advent calendar A Plottl A Day.

Treehugging

Done! Done with Christmas shopping!

My husband just told me he bought a flashlight for my younger brother. Totally ingenious, it has a mode with a red warning signal and can be tied around the head. I'm not entirely sure what my brother is supposed to do with that. But the batteries are allegedly durable for ten years, so maybe something springs into mind.

Either way, I read today this article in the Globe and Mail:

"The hush-hush regreening of Europe"

According to which "[Europe's] forest cover has expanded by almost 10 per cent since 1990, and a much larger greening seems to be under way, reversing centuries of deforestation. The greatest share of this growth is a result of deliberate policies designed to turn farmland into woodland."

Having grown up seeing the forests in my neighborhood shrink every year, it made me very happy reading this. I welcome the trend to support afforestation of farmland not only because I like trees (yeah sure, I talk to my plants), but because the financial support for farmers whose products are not consumed either way and just rot away is nothing but a waste of resources.

That further caused me to look up the websites of the European Environment Agency where one finds a lot of data and statistics, and websites of the Ministerial Conference on the Protection of Forests in Europe (MCPFE), more data and statistics, in my impression trends are overall good. Below you see Fig. 46 from their 2007 Report on the State of Europe's Forests (beware, it's more than 16 MB worth of plots and diagrams). It shows the share of protected area of the forest by country:

[Share of protected area of the total forest and other wooded land area for biodiversity (MCPFE Classes 1.1–1.3) and for landscape (MCPFE Class 2) (%), by country in the MCPFE region, 2005, classified according to the share of forest protected for biodiversity, Source: MCPFE. The * means only data for forest available. Click to enlarge]

I admit on deliberately chosing a figure where Germany looks good. And the pie below shows the share of marketed forest products other than wood... seasons greetings!


[Marketed non-wood forest products from forest and other wooded land in Europe. Share of total value in countries (based on available data). Click to enlarge.]

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This post is not part of our 2007 advent calendar A Plottl A Day.

Winter Solstice

Today is the Winter Solstice, the shortest day of the year in the Northern Hemisphere. At 06:08 UT (1:08 AM EST, 07:08 MEZ), the Sun will reach its southern turning point on the celestial sphere, culminating above some point in the Indian Ocean on the Tropic of the Capricorn. In Frankfurt am Main, Germany, sunrise will be at 08:22, and sunset at 16:26, yielding in principle 8 hours and 4 minutes of sunshine.



Today's plot shows the time of sunrise, Sun's culmination, and sunset for Frankfurt over the course of the year. Date is on the horizontal axis, and time of the day on the vertical axis. The shortest day is marked by the vertical line. Sunrise and sunset are shown in orange, culmination - that is, "the true noon", when the Sun is exactly at South - in red. The lines in black, purple, and blue mark the times of astronomical, nautical, and civil twilight, respectively, when the Sun is 18°, 12°, and 6° below the horizon.

Besides the changes in the length of the day, which is roughly between 8 and 16 hours over the course of the year, the plot shows one more unexpected phenomenon: The time of the culmination of the Sun, the true Noon, is not always at the same time - it varies by about half an hour over the year. As a result, the time span of daylight not only varies in length, but also shifts around slightly with respect to the hours of the day. The difference between true Noon and the time when one would expect the Sun at South from the longitude (the "mean Noon", 12:25 MEZ for Frankurt at 8°41 East, for example) is called the Equation of Time. It is caused by the combined effect of the elliptical shape of the Earth's orbit and the tilt of the Earth's axis.

As a side effect of the Equation of Time, the earliest sunset is not at the Winter Solstice, but a few days before. The figure on the left shows a detail of the plot above: Sunrise, culmination, and sunset between November and February. One can see clearly that there is an offset of about 15 days between the earliest sunset, around mid-December, and the latest sunrise, in the first days of January.

The rapid shift in the time of culmination - about half an hour between beginning of November and end of January - can be understood from the fact that the Earth is at the perihelion, the smallest distance to the Sun on its orbit, around January 3. As described by Kepler's law of areas, encoding the conservation of angular momentum, the Earth's angular velocity as seen from the Sun is biggest at the perihelion. But this means that the Earth has to spin a little bit longer from culmination to culmination than at other phases of the orbit. As a consequence, culmination is always a little late from day to day, and the "true Noon" shifts from a bit later than 12 o'clock to 12:40 between November and February. Summer days, instead, will be a little bit shorter.

But at the meantime, those of us in the Northern Hemisphere who are suffering from the winter's darkness can look forward to the daylight coming back.

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All data for the Sun shown in the plots have been obtained from the Data Service of the U.S. Naval Observatory.

Here is a link to a Daylight Applet that you can use to plot sunset, sunrise and twilight times for any location on the planet.

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This post is part of our 2007 advent calendar A Plottl A Day.

The Cosmological Parameters

[Figure: Supernovae Cosmology Project]

The most puzzling experimental result of the last decade that has inspired hundreds, if not thousands of papers, is the matter composition of our universe. The measurements indicate that the usual matter we are made of is only a small fraction, 4%, of all the matter content of the universe. According to present data, 23% is dark matter, whereas the remaining 73% is dark energy. The matter densities are usually normalized to the so-called critical density ρ_crit = 8 π G/3 H², upon which one obtains the more convenient dimensionless parameters Ω_M for the fraction of matter (us + dark), and Ω_Λ for the dark energy part.

The plot above shows best fit confidence regions in the Ω_Λ versus Ω_M plane. It combines data from supernovae redshift, galaxy clustering, and analysis of the Cosmic Microwave Background. The shaded upper left corner indicates a region where there would be no Big Bang (scale factor doesn't go to zero). The diagonal line is a flat universe, and divides the areas of closed and open models. The slightly upward bended line divides region with and without recollapse (derivative of the scale factor can have a zero). If Ω_Λ = 0, then a closed universe recollapses eventually. Roughly speaking, more matter requires more dark energy to have continuing expansion, so the line dividing continuing expansion from recollapse bends upwards. (Sean Carroll explains you how to compute these boundaries here.)

The shaded lower right corner would imply the the universe was younger than the oldest observed stars. The supernovae results and the CMB data constrain combinations of both parameters, such that the best fit regions lie on diagonal ellipses, whereas the cluster data is dominantly sensitive to the amount of matter, though the ratio of X-ray gas mass to total mass depends on depends on both parameters, so it yields some weak constraints also on Ω_Λ (Allen, Schmidt and Fabian, Mon. Not. Roy. Astron. Soc. 334 (2002) L11 arxiv:astro-ph/0205007).

The present data is compatible with a flat universe. Though one has to be somewhat careful what this analysis actually shows. It shows that the ΛCDM model - a flat cosmology with the above mentioned fractions of dark matter and dark energy - is a parametrization of observed effects that is the best fit to the presently available data. But so far we have no experimental knowledge about the microscopic nature of the unknown constituents of the universe.

Actually, writing this series of posts about today's data and how it is compatible with our theories I am impressed how much the homo sapiens sapiens has learnd about the world around him, ;-).

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This post is part of our 2007 advent calendar A Plottl A Day.

The J/Psi and the Charmonium Spectrum

Atoms emit light at very specific wavelengths - their so-called spectrum is their characteristic fingerprint. The origin of the discrete lines in the spectrum can be understood best for the most simple atom: In the hydrogen atom, which consists of an electron bound to a proton, the electron can exist only with certain distinct energies. Such are the rules of quantum mechanics, and as a consequence, the bound system of the electron and the proton can absorb or set free energy only in specific amounts, which correspond to light with the frequencies, or wavelengths, that show up as lines in the spectrum. The analysis of the spectrum of the hydrogen atom has revealed a lot of details about the electromagnetic interaction between electron and proton, and between electrical charges in general.

[Source: Kay Königsmann: Radiative decays in the Ψ family, Physics Reports 139, Issue 5, June 1986, Pages 243-291, Fig. 5.]

One may wonder, is there a similar phenomenon for bound systems between other particles, say, between quarks, which carry a so-called colour charge and interact via the strong force described by quantum chromodynamics, QCD? The answer is an emphatic yes - and one can learn a lot from it. The figure shows the spectrum of excited states of the J/Ψ meson, a bound charm quark-antiquark pair. Like the electron-proton pair in the hydrogen atom, the quark-antiquark pair can have only specific energies, it can be excited to a series of states with higher energies, and it can emit and absorb light when transiting between these states. The corresponding photon spectrum - the numbers of photons within a small range of energy counted in a detector - is shown in the plot, with a specific pattern of lines superimposed on a smooth background. The numbers help to identify the lines with the transitions between different states, which are shown schematically in the lower part of the figure. Data have been measured at the Crystal Ball, a spherical detector that completely encloses the quark-antiquark pair.

The photons counted in the spectrum do not correspond to visible light, however, but are short-wavelength gamma rays with an energy in the range between 100 and 500 MeV - that's much more than the 1.9 eV corresponding to the red Hα line in the hydrogen spectrum. This should not come as a surprise: The energy scale of the hydrogen atom is set by the Rydberg constant, R = 13.6 eV, which is proportional to α²m, where α ≈ 1/137 is the fine structure constant - the coupling constant of QED - and m is the mass of the electron. Now, a charm quark has about 3000 times more mass than an electron, and the coupling constant of QCD, the strong interaction, is about 100 times bigger than that of QED. Hence, one could expect spectral lines with energies by a factor 30 million times bigger for the charm-anticharm pair than for the hydrogen atom - as you can see, that's not that bad an estimate.

However, there is an essential difference between the hydrogen atom and charmonium, as the bound charm-anticharm pair is usually called, and it can be spotted in the known spectrum of charmonium states, which is shown in this plot.

[Source: Ted Barnes: The XYZs of charmonium at BES, Int. J. Mod. Phys. A21 (2006) 5583-5591 (arXiv: hep-ph/0608103v1), Fig. 1.]

Here, the known charmonium states are shown as black lines according to their energy, or mass, on the vertical axis, and grouped as per orbital angular momentum of the quark-antiquark pair, labelled by S, P, D, F, along the horizontal axis. In contrast to the hydrogen spectrum, there is no series limit at high energies, which in the hydrogen atom corresponds to ionisation, the separation of the electron and the proton. Instead, energy levels increase rather uniformly, but cross a line, labelled "DD". Above that energy, the charmonium system can decay in a D meson, made up of charm quark and a light antiquark, and the respective antiparticle, the anti-D. This is the manifestation of very characteristic feature of QCD, called colour confinement: No isolated quarks, or other colour charges, can be observed. Instead, if one tries to separate, say, the charm quark and the anticharm quark of the J/Psi; by adding energy, a new quark-antiquark pair will be created, and two mesons will be formed, the D/anti-D pair.

Because of confinement, it is clear that the interaction energy between quarks can not be described by a simple analogy to the Coulomb law for electrical charges. To model confinement, it is stipulated that a Coulomb-type interaction has to be amended by some energy which increases linear with charge separation. In fact, one has tired to reverse-engineer an interaction potential between quarks starting from the charmonium spectrum: Making an ansatz for the interaction energy, one can calculate the corresponding spectrum, and fit the parameters to match the observed spectrum. The most popular ansatz is often called Cornell potential and looks like this:

Here, the first term is the Coulomb energy, with the strong coupling constant α_s instead of that of electrodynamics, the second term is the energy linear in distance, which enforces confinement, and the third term contains spin-dependent terms to model the fine structure of the spectrum. The constant κ is the the so-called string tension. In the spectrum shown in the figure, the best fit to the experimental data with this spectrum is shown in red. It corresponds to a mass of the charm quark of m_c = 1.46 GeV/c², a coupling constant α_s = 0.55, and a string tension (called "b" in the plot) of κ = 0.72 GeV/fm - meaning that an energy of 0.72 GeV is needed to separate the quark-antiquark pair by 10^-15 meters.

Understanding confinement of colour charge is an open problem in physics, and the details of the interaction between quarks forming a hadron still contain many riddles. The analysis of the spectrum of quark-antiquark pairs as in charmonium can help to a better understanding of these issues - that's why the spectrum of charmonium is still an active area of research.

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The best electrodynamcial analogue to charmonium is not the hydrogen atom, but the bound state of an electron and a positron, called positronium. The nomenclature of charmonia, and the name charmonium itself, derives from positronium physics.

The Cornell potential got his name after the group of physicists at Cornell who had used it within weeks of the discovery of the J/Ψ to analyse the spectrum of its excited states - see E. Eichten et al.: Spectrum of Charmed Quark-Antiquark Bound States, Phys. Rev. Lett. 34 (1975) 369.

Charmonium physics will be one main topic of the PANDA (Proton ANtiproton DArmstadt) experiment at the new "Facility of Antiproton and Ion Research" (FAIR) at GSI, Darmstadt, Germany (see, e.g., Bertram Kopf: Physics with Antiprotons at PANDA, J. Phys.: Conf. Ser. 69 012026).

The Crystal Ball Detector is now in use at the Mainzer Mikrotron (MAMI), Germany.

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This post is part of our 2007 advent calendar A Plottl A Day.

Indirect Detection of Gravitational Radiation

[Source: J.M. Weisberg, J.H. Taylor: Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis, arXiv: astro-ph/0407149v1 Figure 1.]

One of the predictions of Einstein's General Theory of Relativity is the existence of gravitational waves: Two large masses in orbital motion will create small, wavelike distortions in spacetime that propagate like ripples on a pond, and carry away energy.

Despite big efforts and huge detectors, no gravitational waves could be measured so far. But direct discovery notwithstanding, physicists are confident that gravitational waves are real, since there is very compelling indirect evidence for their existence, and for their compliance to the rules of General Relativity.

A typical source of gravitational waves would be a couple of very massive, compact stars in close orbits. Binary neutron stars are good candidates for sources of gravitational radiation. The energy oss by the emission of gravitational waves would result in a slowly decaying orbit: the stars would et ever closer the more energy would be radiated away as gravitational waves. And if one of the stars is a pulsar, there are chances that one can measure the orbit, and establish the indirect consequences of the emission of gravitational waves.

That is exactly what has happend with the binary pulsar Binary Pulsar PSR 1913+16: In December 1973, astronomy student Russell A. Hulse was at the Arecibo Radio Observatory in Puerto Rico, collecting data on pulsars for his Ph.D. thesis. One of the pulsars he was observing showed a curious, periodic variation in the pulsation frequency. It soon became clear that this variation could be best understood as the periodic Doppler shift of the pulsar in an elliptic, Keplerian orbit around another star. This allowed for the very precise reconstruction of the orbit of the pulsar.


Schematic picture of the binary pulsar PSR1913+16: Two neutron stars circle each other closely in elliptical orbits. Their mean separation is only a few times the distance Earth-Moon, and one orbit takes only about eight hours. When the stars pass close to each other, they emit large amounts of gravitational radiation. [Source: Nobel Foundation, 1993.]

The orbital parameters of this binary system, however, are so extreme - masses on the order of the Sun orbiting within eight hours at 1 permille of the speed of light at a distance on the order of the distance Earth-Moon - that Kepler's laws for the motion of two masses under the influence of gravitation are not sufficent anymore: As Hulse's advisor Joseph Taylor noted, the full-scale apparatus of General Relativity was necessary to describe the orbit, instead of the simple Newton law of gravitation, For example, the binary pulsar shows a big motion of the periastron - the equivalent of the perhelion shift of Mercury, whose share not accounted for by the perturbations of the other planets in the Solar System was explained by Einstein as the first "postdiction" of his new theory of gravitation. What's more, the time evolution of the orbit can be measured precise enough to check for the consequences of the emission of gravitational waves!

The data points in figure show the observed change in time of periastron over the last 30 years, since the first data on pulsar PSR 1913+16 have been available. The parabola illustrates the theoretically expected change in periastron time for a system emitting gravitational radiation, according to general relativity.

Not only the Nobel committee in Stockholm considered this an extremely important result - but they awared to Russell A. Hulse and Joseph H. Taylor, Jr., the Nobel Prize in Physics 1993 for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation.

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Taylor's Nobel Lecture is on Binary Pulsars and Relativistic Gravity (PDF file), while Hulse in his Nobel Lecture talks about The Discovery of the Binary Pulsar (PDF file).

The Binary Pulsar PSR 1913+16 is a nice short introduction that explains in more detail how the pulsar data are analysed, and how the periastron shift can be extracted.

For the latest data about the "Taylor-Hulse" binary pulsar PSR1913+16, check out Wikipedia.

For two recent related rviews, see Duncan R. Lorimer: Binary and Millisecond Pulsars, Living Rev. Relativity 8, (2005), and Ingrid H. Stairs: Testing General Relativity with Pulsar Timing, Living Rev. Relativity 6, (2003).

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This post is part of our 2007 advent calendar A Plottl A Day.

Neutrino Masses and Angles

Neutrinos come in three known flavors. These flavors correspond to the three charged leptons, the electron, the muon and the tau. The neutrino flavors can change during the neutrino's travel, and one flavor can be converted into another. This happens periodically. The neutrino flavor oscillations have a certain wavelength, and an amplitude which sets the probability of the change to happen. The amplitude is usually quantified in a mixing angle θ. sin²(2 θ) = 1, or θ = π/4 corresponds to maximal mixing, which means one flavor changes completely into another, and then back. For a brief introduction, see also our earlier post Neutrinos for Beginners.

This neutrino mixing happens when the mass-eigenstates of the Hamiltonian are not the same as the flavor eigenstates. The wavelength λ of the oscillation turns out to depend (in the relativistic limit) on the difference in the squared masses Δm² (not the square of the difference!) and the neutrino's energy E as λ = 4E/Δm². The larger the energy of the neutrinos the larger the wavelength. For a source with a spectrum of different energies around some mean value, one has a superposition of various wavelengths. On distances larger than the typical oscillation length corresponding to the mean energy, this will average out the oscillation.

The plot below from the KamLAND Collaboration shows an example of an experiment to test neutrino flavor conversion. The KamLAND neutrino sources are several Japanese nuclear reactors that emit electron anti-neutrinos with a very well known energy and power spectrum, that has a mean value around some MeV. The average distance to the reactors is ~180 km. The plot shows the ratio of the observed electron anti-neutrinos to the expected number without oscillations. The KamLAND result is the red dot. The other data points were earlier experiments in other locations that did not find a drop. The dotted line is the best fit to this data.



[Figure: KamLAND Collaboration]

One sees however that there is some kind of redundancy in this fit, since one can shift around the wavelength and stay within the errorbars. These reactor data however are only one of the measurements of neutrino oscillations that have been made during the last decades. There are a lot of other experiments that have measured deficites in the expected solar and atmospheric neutrino flux. Especially important in this regard was the SNO data that confirmed that indeed not only there were less solar electron neutrinos than expected, but that they actually showed up in the detector with a different flavor, and the KamLAND analysis of the energy spectrum that clearly favors oscillation over decay.

The plot below depicts all the currently available data for electron neutrino oscillations, which places the mass-square around 8×10^-5 eV², and θ at about 33.9° (i.e. the mixing is with high confidence not maximal).




[Figure: Hitoshi Murayama, see here for references on the used data]

The lines on the top indicate excluded regions from earlier experiments, the filled regions are allowed values. You see the KamLAND 95%CL area in red, and SNO in brown. The remaining island in the overlap is pretty much constrained by now. Given that neutrinos are so elusive particles, and this mass scale is incredibly tiny, I am always impressed by the precision of these experiments!

To fit the oscillations between all the known three neutrino flavors, one needs three mixing angles, and two mass differences (the overall mass scale factors out and does not enter, neutrino oscillations thus are not sensitive to the total neutrino masses). All the presently available data has allowed us to tightly constrain the mixing angles and mass squares. The only outsider (that was thus excluded from the global fits) is famously LSND (see also the above plot), so MiniBooNE was designed to check on their results. For more info on MiniBooNE, see Heather Ray's excellent post at CV.

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This post is part of our 2007 advent calendar A Plottl A Day.

PS on "ain't it thrilling"

As a PS to yesterday's post, a photo regarding the signs buried under snow:




Taken 1 hour ago in the PI parking lot.

Three Neutrino Families

[Source: The ALEPH Collaboration et al., Precision Electroweak Measurements on the Z Resonance, Physics Reports 427 (2006) 257; arXiv: hep-ex/0509008v3, Fig. 1.13.]

In my car, parts of the glove compartment start to make a rattling noise at a specific engine speed. That's a typical example of a resonance: A mechanical system that can sustain vibrations of a certain frequency, called the natural frequency, will start to vibrate if it is driven by an external excitation with a frequency that closely matches the natural frequency. If there is damping in the vibrating system, the response to the external excitation is smaller than without damping, but on the other hand, response sets in already for larger mismatches between driving and natural frequency than without damping.

In a very similar way, the annihilation of electrons and positrons in a particle collider can produce a new particle if the centre-of-mass energy of the electron-positron pair matches the energy corresponding to the mass of the particle. Most particles produced in this way - for example, the J/Ψ and Υ mesons, or the electrically neutral Z boson, the massive partner of the photon in the electroweak theory, - are unstable and will decay on a very short time scale into other particles. For example, the Z particle can decay in pairs of charged leptons (electron and positron, muon and antimuon, tau and antitau), in neutrino-antineutrino pairs (one pair for each charged lepton pair), or in quark-antiquark pairs, which will end up in some hadrons. These processes can be represented by the diagrams

or or .

As a result, the Z boson, with a mass of 91.2 GeV/c², decays with a half-life of about 10^-25 seconds into any of these particle pairs. But this means, in the language of resonances, that the Z is quite strongly damped, and that it can be produced also if there is a certain mismatch between the mass of the Z and the centre-of mass energy of the electron-positron pair.

This is shown in the green curve in the figure, which fits the data measured at different experiments at the now dismantled Large Electron Positron Collider (LEP) at CERN. The curve shows the total cross-section σ_had for the production of hadrons as a function of the center of mass energy E_cm. This cross-section has a clear maximum at the energy corresponding to the mass of the Z, but this is not a sharp peak - the bump is quite broad, and has a certain width Γ of about 2.5 GeV. This width is related to the lifetime τ of the Z by the relation Γ·τ = ħ, where ħ is Planck's constant divided by 2π.

But there is more information hidden in the resonance curve of the Z: From all the different decay products, charged lepton pairs, such as muon-antimuon pairs, and hadrons can be measured in detectors, but the neutrino-antineutrino pairs are elusive. How can one be sure that they are there, and can one say anything about their number?

The exciting point is that one can see that they are there, and that there are three different families of neutrinos: the electron-, muon-, and tau neutrino with their respective antineutrinos.

In fact, while not directly detectable, the neutrinos contribute to the damping of the Z boson, and hence, to the height and width of the resonance peak. For example, one can calculate how the resonance curve should look like if there were only two neutrino families, or four neutrino families: The Z would be less, or more damped than for three families, and the peak should be higher and sharper, or flatter and wider, respectively. The corresponding curves are shown in the plot, labelled as "2ν" and "4ν", respectively.

Data fit perfectly well the resonance curve for three neutrino families. Taking into account the caveat that a neutrino has to couple to the Z (it should not be "sterile"), and that its mass should be below half the mass of the Z (the Z should be able to decay into the particle-antiparticle pair) for the method to work, these LEP data show that there are three neutrino families - 2.9840 ± 0.0082, to be precise...

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All the details from The ALEPH Collaboration, the DELPHI Collaboration, the L3 Collaboration, the OPAL Collaboration, the SLD Collaboration, the LEP Electroweak Working Group, the SLD electroweak, heavy flavour groups in Precision Electroweak Measurements on the Z Resonance, Physics Reports 427 (2006) 257; arXiv: hep-ex/0509008v3

OPAL Events at LEP1 is a beautiful collection of event displays that shows how the different processes that are involved in the creation and decay of the Z boson look like in the particle detector.

The form of the resonance peak is known as Breit-Wigner distribution. The width Γ is the full width at half maximum (FWHM).

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This post is part of our 2007 advent calendar A Plottl A Day.

ain't it thrilling

It's this time of the year. This time when I get emails from friends and relatives who I haven't heard of the last 12 months. Do you have snow, they ask. Well, yes, we do have snow. Let me tell you how that looks like here.

When it snows, ain't it thrilling,
Though your nose gets a chilling

I try to leave the house in the late morning but find I can't open the door, since the wind has pushed up half a meter snow in front of it.

So I leave the house through the back door just to realize that it's still snowing rather heavily, and I either can't see where I'm walking because my glasses are all snowy, or I can't see where I'm walking because I'm not wearing the glasses. Fighting with the glasses I drop the house-key on what I think should be lawn somewhere below the layers.

I spend 10 minutes searching the stupid house-key, just to then find that it was actually not the house-key I took with me, so I have to go around the house back to the front door.

Where it dawns to me that the reason why the walkway isn't cleared is that my landlord is on vacation since yesterday. And since I haven't really socialized with the neighbors, this means there's nobody who will let me into the house.

I shovel away the snow enough to squeeze myself into the hallway, and then spend another 10 minutes trying to convince an elderly lady that I am neither trying to rob her nor will I sleep in the lobby if she lets me in. I convince her with proving the key I took with me fits to the mailbox, which reveals 3 weeks worth of Pizza delivery flyers.

I then find it would be a better idea to take the car. My timing turns out to be excellent because somebody is just leaving directly in front of me. That somebody courageously makes it down the driveway, where his car gets stuck with the headlights halfways buried in snow where the sideway ends.

It takes 15 minutes to dig out the neighbors car and shove it onto the road. After which I start up high speed and hop on the road as well. The radio announces an endless list of closures and cancellations, and informs me the government of Ontario recommends to drive only in case of emergencies until the roads are cleared.

Sleigh bells ring, are you listening,
In the lane, snow is glistening

A couple of cars slide around, make 360° turns on the street crossings, or spray snow fountains while unsuccessfully trying to get through the tougher parts of the road.

I consider getting winter tires, what do you think?

People walk on the streets because the sidewalks are a disaster. Dogs dig their noses into the snow where other dogs have pissed holes. In rare places snow has been piled up already on the hills from last week, that easily exceed several meters height. Some street signs get buried in these hills. And folks, this is just the beginning of the Winter fun!

Later on, we'll conspire,
As we dream by the fire
To face unafraid,
The plans that we've made

I think you get the picture? Let me just add the above describes a rather average winter day. For me this year is a definite improvement over last year, because I now have a garage to park the car in, so I won't have to enter through the trunk if the doors are frozen shut.

Some other things I've learned last winter:

• Get the Coke out of the car before it gets really cold and it bursts, thereby splashing frozen Coke chunks all through the car's interior.
• If you didn't get the damned Coke out of the car in a timely manner, scrape off the frozen Coke chunks before Spring time.
• Don't wear shoes with laces. They will become completely stiff after two days because of all the salt on the streets.
• Your CD player, Ipod, Digital Camera, they all have a minimum working temperature below which they will refuse to function. I recommend pairing the Ipod with one of these Chemical Hand warmers. (No idea though why the guy says there is something new about it, I've had them since I was a kid.)
• Consider wearing socks with your Flip Flops.
• Dry your hair before leaving the house. Dry it thoroughly.
• Softness of chewing gum depends crucially on the temperature.
• If your stupid sliding window doesn't open, better leave it closed. If you'll try to unfreeze the ice, chances are you won't be able to close it again which is much worse.
• Don't cry if the outside temperature is below - 25 ° C.

Gravitational Microlensing for Detection of Extrasolar Planets

Einstein taught us mass curves space-time, and everything moving in it has to behave accordingly, even light. Obeying the laws of General Relativity, large masses bend light rays around them. Essentially a mass acts like a convex lens, and collects light rays that otherwise would have missed the focal point - here the observer, an Earth or space based detector. This effect is called gravitational lensing. One speaks of gravitational microlensing if the object causing the light bending is of solar size. (If you have the Windows Media player, check out this nice animation from the NSF.)

If the lensing object crosses our line of sight to a more distant source star, it will affect the light from that star, producing two or more close images whose total brightness and magnification is enhanced. If the lensing star is accompanied by a planet, one can potentially observe not only the lensing effect from the star, but also a smaller effect resulting from the presence of the planet.

The plot below shows a particularly nice example, the detection of an extrasolar planet with the poetic name OGLE-2005-BLG-390Lb. Observed in July 2005, it was estimated to have a mass about 5.5 times that of the Earth. The plot shows the magnification of the source object, a bright G2 giant, due to the crossing of the star OGLE-2005-BLG-390. The small bump is the effect of the planet. The data set consists of 650 data points from various observatories, the errorbars are 1 σ.


[Reprinted by permission from Macmillan Publishers Ltd: Nature 439 437-440 (26 January 2006) doi:10.1038/nature04441, Copyright 2005.]

The top left inset shows the light curve of the previous 4 years, and the top right one shows a zoom of the planetary deviation on a time interval of 1.5 days. The solid curve is the best fit with the star and planet system. The dashed grey curve is the fit with best binary source model (two independent lensing stars) that is rejected by the data, and the dashed orange line is the best single lens model. These results were published in Nature 439, 437-440 (26 January 2006).

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News report from the Probing Lensing Anomalies NETwork (PLANET): Discovery of OGLE 2005-BLG-390Lb, the first cool rocky/icy exoplanet.

Paper by J.-P. Beaulieu et al.: Discovery of a cool planet of 5.5 Earth masses through gravitational microlensing, Nature 439, 437-440 (26 January 2006) doi:10.1038/nature04441

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This post is part of our 2007 advent calendar A Plottl A Day.

Posts per Second

As to April 2007, Technorati counted 17 posts per second on blogs they track. The plot below shows the number of tracked blogs from March 2003 on.


[Plot: Sifry, Click to Enlarge]


By April 2007, they counted about 70 million weblogs, and 120,000 new weblogs each day. The statistic below shows the number of daily posts between Aug 2004 and Feb 2007


[Plot: Sifry, Click to Enlarge]

For more statistics on the blogosphere, see The State of the Live Web.

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This post is part of our 2007 advent calendar A Plottl A Day.

Asymptotic Freedom and the Coupling Constant of QCD

[Source: Siegfried Bethke, Experimental Tests of Asymptotic Freedom, arXiv:hep-ex/0606035, Fig. 17.]

Life without interaction is boring. If quarks would not interact, there would be no protons, no neutrons, no atoms, no readers of blogs. In quantum chromodynamics (QCD), the theory that in principle explains how quarks bind to protons, this interaction between quarks is described by the exchange of gluons - particles that glue together the quarks. The very naive idea is the following: two quarks exchange a gluon with momentum Q, and depending on the colour charge of the quarks, this exchange results in an attraction or a repulsion between both quarks.


The strength of the interaction depends of a factor which is called the "coupling constant", which for quarks and gluons is usually denoted as α_s. Here, the index "s" stands for "strong", since the interaction between quarks and gluons had been called the "strong interaction" for reasons that show up nicely in the plot above, as we will see in a second. The exchange of one gluon is proportional to a factor g² = 4πα_s - in the diagram, each on of the two vertices where the gluon and the quark get in touch contributes a factor of g, the square root of 4πα_s.

This is completely analogous to quantum electrodynamics, where the exchange of a photon between two electrons is proportional to e², the product of the electrical charges of the two interacting particles - the very same factor that has been known since a long time from Coulomb's law for the force between charges. In electrodynamics, the constant α = e²/4π is called the fine structure constant - it's a pure number, without dimensions of length or mass, and has the value α ≈ 1/137. Moreover, it has the nice property to be more or less independent of the momentum Q of the photon that is exchanged. The smallness and the constancy of α in QED allow all kinds of calculations that are in pretty good agreement with experiment.

In QCD, alas, things are more complicated, and the main reason for this is encoded in the plot above. It shows a compilation of the values for α_s, derived from many different experiments, and for different momenta Q of the exchanged gluons. Gluon momentum is measured in GeV/c (and c, the speed of light, is set to 1), and a logarithmic scale has been used to allow to show a bigger range of values.

There are two features of the curve which correspond to two main characteristics of quantum chromodynamics:

The "coupling constant" α_s is not a constant at all - it decreases with increasing momentum. Moreover, it lies in the range 0.1 - 0.3 at values of Q that can be probed in experiment, which means that it's about 50 times larger than the fine structure constant of electrodynamics - that's why the "strong interactions" are strong -, and the factor g² = 4πα_s is on the order of 1, and bigger than 1 for small momenta.

This second feature is called "asymptotic freedom", and it means that quarks are nearly free, or non-interacting, when the exchange momentum is very big. As a result, the computational tools which are so successful for electrons and photons can be applied to quarks and gluons at very high energies.

The other side of the coin, however, is that phenomena at lower energies are much harder to calculate. And, for example, in the regime where quarks bind together to protons or other hadrons, α_s is too big to use the recipes of quantum electrodynamics. So far, there are only numerical methods available to solve the full equations of QCD for hadrons, and many different analytic approximation schemes.

Which makes, on the other hand, the question of how quarks interact to build a proton as challenging as interesting.

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More than you ever wanted to know about the running coupling of QCD can you find, e.g., in the paper by Siegfried Bethke: Experimental Tests of Asymptotic Freedom, arXiv:hep-ex/0606035, and Progress in Particle and Nuclear Physics 58 (2007) 351-386, and in the review Quantum Chromodynamics and its coupling by I. Hinchliffe for the Particle Data Group (PDF file).

On asymptotic freedom, and QCD in general, you can check out QCD Made Simple (Physics Today) and Asymptotic Freedom: From Paradox to Paradigm (arXiv: hep-ph/0502113) by Frank Wilczek, who together with David Gross and David Politzer was awarded the Nobel Prize in Physics 2004 for the discovery of asymptotic freedom in QCD.

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This post is part of our 2007 advent calendar A Plottl A Day.

The Photoelectric Effect

A couple of years ago I tried to get a group of undergrads excited about the photoelectric effect. Some of them got so excited they fell asleep. Others built impressive constructs with roller pens. A few typed away on their cell phones.

When I was through with my exciting lecture one of them looked up from the display, and asked me what it's good for. Well, to show that the energy of the light is proportion to the frequency, I explained pa-ti-ent-ly. Ah, he said, but isn't the frequency of light the same as the energy?

See, that's what happens if ħ is equal to one in textbooks from the school level on. But more seriously, I figured the students had just learned from the very beginning on that frequency is essentially the same as energy. So then what's the big deal with the photoelectric effect? And why on earth did somebody get a Nobel Prize for it?

Well, until the last century students didn't have cellphones, h didn't have a bar, and light was a wave. A wave has an amplitude and a frequency. If you turn up the volume of your stereo its the amplitude of the sound waves that you change, not the frequency. If you turn the dimmer of your living room light, it's the light's amplitude that you change, not the frequency*.

In 1899 Thomson established that ultraviolet light caused electrons to be emitted from a metal surface. This was believed to be due to the atoms being shaken around by the infalling light waves, such that an electron could escape. In this case however, a higher light intensity should result in more emitted electrons and with more intensity the electrons should have a larger (average) kinetic energy.

So it came as a surprise what von Lenard found in 1902 when he studied how the energy of the emitted electrons varied with the intensity of the light. For this, he placed a negatively charged plate, the collector, opposite to the plate on which the light fell. The electrons that were emitted were repelled by the plate, and could only reach it if they had sufficiently high kinetic energy. If they reached the plate, they would cause a current that was measured.

Lenard found that there was a minimum voltage V_stop at the collector at which no electrons would reach it. The expectation was that increasing the light's intensity would then equip the electrons with more kinetic energy, and thus raise the repelling voltage necessary to stop them from reaching the collector. But it turned out V_stop did not depend on the intensity of the light. Instead it varied with the light's frequency.

In 1905 Einstein explained these findings by suggesting that the light should be thought of as quanta of frequency hf, with f the frequency that kick out the electrons from the plate. The electron would then carry the light quanta's energy, minus some constant energy that needs to be provided to get the electron off the metal surface. If the voltage is adjusted such that it stops the electrons from reaching the collector, e V_stop should be linear in the light's frequency with the constant of proportionality being Planck's constant. The plot below shows this dependence. On the y-axis you see the stopping voltage; the x-axis shows the frequency of the infalling light. The box in the corner is the computation of the curve's slope which gives Planck's constant.

Source: Robert A. Millikan's Nobel Lecture The Electron and the Light-Quant from the Experimental Point of View, The Nobel Foundation, 1923.

A. Einstein received the Nobel Prize in 1921 for "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect", and R.A. Millikan received the Nobel Prize two years later "for his work on the elementary charge of electricity and on the photoelectric effect".

And now you can set ħ = 1 again.

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* Roughly speaking. I guess the spectrum of the emitted frequencies depends somewhat on the voltage.

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This post is part of our 2007 advent calendar A Plottl A Day.

The Hadron-Muon Branching Ratio

Today's plot is my favourite plot from high-energy physics - it's a compilation of data measured at diverse particle colliders that shows what happens when an electron annihilates with its antiparticle, the positron, at very high energies.

Source: Particle Data Group, Plots of cross sections and related quantities, Fig. 6 (PDF file).

As the production of the J/Ψ resonance has already shown us, different things can happen in such a collision: The electron and the positron can just bounce off each other, a process called Bhabha scattering, or they can annihilate, and their energy, if high enough, can materialise in new particles. The most abundant products of this materialisation processes are muon-antimuon pairs (the muon, μ^-, is a massive cousin of the electron), or one or more hadrons, such as pions, which are made of quark-antiquark pairs.

The probability of these different possible events is measured by the so-called cross-section σ: one imagines the particles rushing onto each other as small disks - the larger the area of the disk, the higher the probability that two particles will hit each other and something will happen. The area of these imaginary disks is the cross section.

The figure shows the ratio R of the cross-sections for the creation of hadrons to the creation of muon-antimuon pairs in electron-positron collisions as a function of the centre-of-mass energy of the electron-positron pair, called "square root of s" for historical reasons. Both energy and ratio R are plotted on a logarithmic scale to allow the representation of a larger range of values. Energy is measured in Gigaelectronvolt (GeV, that's roughly the energy equivalent of the mass of a hydrogen atom), and the ratio is given more precisely by


The diagrams on the right-hand side show symbolically what happens in the collision: electron and positron meet and annihilate into a so-called virtual photon γ*, which then materialises as either a muon-antimuon pair or a quark-antiquark pair. But they are not just symbolic: there are very precise rules to convert these Feynman diagrams into actual numbers for the cross-sections σ. Of course, no free quarks have ever been seen in particle detectors, so the creation of a quark-antiquark pairs will be followed by some process that converts them into hadrons. The details of this process are not completely clear yet, but fortunately, the cross-sections entering the ratio R can be calculated without detailed knowledge about the hadronisation process.

Now, there are a few very interesting features about this plot.

First, there are the quite broad peaks shown in blue at relative low energies. These peaks correspond to the creation of mesons made up of light quark-antiquark pairs: the ρ, ρ' and ω of up-antiup and down-antidown pairs, the φ of strange-antistrange pairs. Then follow, as as shown in red, some very sharp spikes: Here, charm-anticharm pairs are created, which hadronise as J/Ψ and Ψ' particles, and at higher even energies, bottom-antibottom pairs materialise, which form the so-called Upsilon Υ and its excited states. From these particles, one can learn a lot about the bound states of quarks and antiquarks and the forces acting between them.

Second, if one takes a closer look, one can see that the flat parts of the curve between the ρ' and J/Ψ spikes, between the J/Ψ and the Υ, and following the Υ are increasing in small steps. This stepwise increase is not difficult to understand: When the energy of the collision becomes high enough that, say, charm-anticharm pairs can be produced, there is a new channel opening for the production of hadrons, while the production of muon-antimuon pairs is not affected. But there is more to learn from these steps: They show that quarks come in three different colours!

In fact, a quite precise first approximation to the ratio R (from a so-called "tree-level" calculation) shows that R is given by the sum of the squares of the charges of the different quarks that can be produced. This estimate is by a factor of three below the experimental data... unless, of course, one takes into account that each quark can come in three different colours!

Third, at the upper end of the energy scale at about 90 GeV, there is a further peak, the so-called Z pole. No new quark-antiquark pairs are created at this peak, but the Z boson, the massive partner of the photon in the electroweak theory. At this pole, the annihilation of the electron-positron pair can happen not only via a virtual photon, but also via a virtual Z boson, and both possibilities have to be added:

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The Z pole contains a very interesting piece of information about the standard model of particle physics, but that will be the story of another plot.

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This post is part of our 2007 advent calendar A Plottl A Day.