Update: On Saturday, I added a blog entry discussing possible mistakes in the Opera paper.
Update: On Thursday, AP, BBC (who renamed Opera to Orion: fixed) and Reuters (and all other media) finally leaked the news: the neutrinos are claimed to have arrived 60 nanoseconds before the light (18 meters over c).
Because this is said to be a 6-sigma signal, their total error margin of the timing should be 10 nanoseconds (3 meters over c); recall that the distance is 732 km. I leave it to the reader to decide whether this accuracy is plausible given the messy birth and detection of the particles. One nanosecond is the duration of one cycle of your iPhone microprocessor, among other things. Ten nanoseconds is 40% of the lifetime of the charged pion or 80% of the lifetime of the charged kaon. I can kind of imagine that they're doing something really silly, like imagining that each pion or kaon lives at least for the lifetime and then it dies. But some of them decay immediately; this error could erase most of the 60-nanosecond discrepancy.
Text below was originally posted on September 19th
Experimental claims on neutrinos that are faster than light
Tommaso Dorigo is spreading a six-sigma (6.1, to be more precise) rumor involving neutrinos and his countrymates in an experiment melodically named Opera (well, the result is actually mostly due to French intruders from Lyon, but that's a detail):
When we visited Milan, a pigeon managed to deposit some products of its metabolism directly into the pocket of my shorts. This procedure was apparently achieved from a nearly impossible (vertical?) angle – much like if your world line in the spacetime were spacelike instead of timelike.
This experience taught me to be careful about angles measured in Italian towns, especially the towns that are famous for an opera. That lesson will turn out to be useful momentarily – in the case of the spacetime surrounding the Gran Sasso mountain.
A theater that is prettier than La Scala.
And by the way, we were so disappointed by the exteriors of the famous La Scala, the local opera house, that I decided to post a picture of our local J. K. Tyl's Theater in Pilsen which all Pilsner tourists preferred over La Scala. ;-)
Make no mistake about it, this trash talk is a preparation for the soccer match, FC Viktoria Pilsen vs AC Milan, in the UEFA Champions League. We will show them the door much like Napoli did a few days ago ;-) before we will do the same thing with Barcelona, a Catalan village where they thought they could play soccer before they met any players from Pilsen. :-) Unfortunately, BATE Borisov (Belarus) played a tie against Pilsen in the first match of our group; Milan tied with Barcelona in our group, too.
Back to neutrinos
Fine. What Dorigo says is simple. They measured how much time it takes for the tau neutrinos to cause a reaction in Gran Sasso tavern in central Italy. The (currently) tau neutrino \(\nu_\tau\) came together with a mostly muon-neutrino beam originally created by decays of pions \(\pi\) and kaons \(K\) into muons \(\mu\) and muon neutrinos \(\nu_\mu\) at CERN, more precisely in SpS which is a pre-accelerator of the LHC.
They tried to be careful about the timing, measured the delay, calculated the speed. And their resulting speed of the neutrinos is
\[ v = c + 6 \sigma. \] It is greater than the speed of light – by an excess of speed that is six times larger than their total error margin. If this claim were true, the probability that this excess happens by chance is lower than one part in half a billion. Much like the pigeon's feces near the opera house in Milan, the Opera collaboration claims that the neutrinos are moving along impossible angles in the spacetime.
The Opera experiment
Of course, the claim is false although I can't tell you exactly why because I wasn't told any further details about the rumor, either. They have surely screwed the calculation of some timing in the multi-step process.
The claims about a superluminal neutrino have a long history. All those claims in the past seem to be impossible although people are usually unaware of the new evidence that identified the errors in the superluminal claims.
Supernova observations are the most reliable source to tell us that neutrinos are moving slower than the speed of light – because their mass is actually positive. For high energy neutrinos, the deviation of the speed from the speed of light is smaller and the neutrinos arrive simultaneously with the light produced during the supernova explosion. But high-energy neutrinos that are faster than light seem to be incompatible with some observations of the supernovae.
You have to assume that neutrinos have a different speed when they move through the Italian rocks. They're perhaps faster in rocks than in the vacuum, Dorigo will tell you. But rocks are almost transparent for neutrinos, according to everything we know about neutrinos today. It's hard to imagine how the rocks could matter. Moreover, local interactions of the rocks with the neutrinos, if they were relevant, would almost certainly slow the neutrinos down instead of speeding them up.
More importantly, superluminal neutrinos – or any other particles – contradict special relativity. If you could shoot superluminal neutrinos, they would move along spacelike paths in the spacetime and the Lorentz symmetry would allow you to boost the SpS a little bit so that the neutrinos would be moving backwards in time (there is no qualitative difference between spacelike paths going to the future and those going to the past). By boosting the SpS accelerator, you could create neutrinos that would move backwards in time and you could kill your grandfather before he had his first sex with your grandmother, thus rendering your own existence needed for the homicide inconsistent with the result of the homicide.
This shouldn't happen. Your grandfather in the past has the right to protect his record of having sex with your grandmother – much like the Italian prime minister has the right to defend his record that he was able to do eight women in a single night (he couldn't manage more, he says, even though eleven women, one for each dimension of the spacetime in M-theory, were waiting in the line). So neutrinos should better be slower than light, otherwise the pride of Berlusconi who just left a court (in Milan, of course) would be in trouble. And the pride matters more than his job because he's only a prime minister in his spare time, we were told. :-)
CERN and Gran Sasso are 732 km i.e. 2.44 light milliseconds away from each other.
More than ten years ago, around the year 2000 or so (when I was a Rutgers grad student), there were claims that the radioactive decay of the tritium atoms implied that neutrinos should have been superluminal. The statistical significance of the claims was apparently very low. I don't know what exactly happened with that claim but because no one has talked about this revolution in recent years, I guess that it slowly evaporated. This was the first time when people tried to quantitatively analyze the beta-decay of atoms and not just the bare nuclei so I am sure that they did something wrong, or they were just unlucky with the statistics.
Tachyons and spin
In bosonic string theory, the 26-dimensional ancestor of modern (super)string theory that was randomly discovered in the late 1960s, one has a troublesome particle called the tachyon. It is a scalar particle associated with the field \(T(x^\mu)\) whose squared mass is negative:
\[ m_T^2 = -\frac{1}{\alpha'} = -\frac{T_{\rm string\,\,tension}}{2\pi} \] for the open string tachyon and 4 times larger for the closed string tachyon. At the level of classical particles, the negative squared mass looks like faster-than-light propagation of particles that allows you to inconsistently change your grandfather's bedroom record.
At the level of field theory, a negative squared mass means that the potential (density of potential energy to be more accurate),
\[ V(T) = \frac{m_T^2}{2} T^2, \] acquires a maximum rather than minimum at \(T=0\). That's very bad because this value of the tachyon field becomes unstable. The field wants to roll down and find a true minimum. If it doesn't find a true minimum, it will continue to roll indefinitely. Quantum mechanically, this increasingly fast rolling on the slope looks like a spontaneous creation of tachyonic particles or their pairs with both positive and negative energies.
Superstring theory eliminates tachyons and guarantees that the theory contains fermions as well. But we have never seen a string theory with a tachyon which is not a scalar particle. Is it a coincidence? Can't we have a tachyon with spin? A Dirac tachyon? Or a vector tachyon?
The answer is that it is no coincidence. In string theory, tachyons have to be scalar particles. The reason is that to give the tachyon a nonzero spin, you must excite it by some stringy oscillators that are connected with large spacetime coordinates. But these oscillations raise the squared mass of the resulting particle and because of that, the particle's squared mass can no longer be negative. In bosonic string theory, you may only obtain spin by \(\alpha_{-n}^\mu\) which increases \(N\) by one (minimal amount for \(n=1\)); in supersymmetric string theory, one may also have \(\psi_{-n}^\mu\) where \(n\) may be as low as \(n=-1/2\) in the antiperiodic (NS) sector. However, the ground state energy is less tachyonic in the RNS superstring case and this is enough to bring you to the massless level with just a single excitation, too.
Of course, those calculations only apply to the free string theory – without any interactions. (But the result is still true even in you consider arbitrary orbifolds etc., even in the twisted sectors where the ground state energy is less tachyonic or not tachyonic at all.) Can you have Dirac particles when you turn on interactions? Can you have spin-1/2 fermions which are tachyonic?
Effective field theory vs spinning tachyons
Well, the answer is that this is impossible even at the level of effective field theory. Recall that the kinetic term for a Dirac particle is
\[ {\mathcal L}_{\rm kin} = \bar \psi \gamma^\mu \partial_\mu \psi \] and is independent of the mass. The mass term
\[ {\mathcal L}_{\rm mass} = -m \bar\psi \psi \] has a coefficient of \(m\). Note that the mass is not squared in this case (unlike the case of the scalar tachyonic particle). Consequently, you can't make the world lines spacelike because that would mean an imaginary value of \(m\) and the action would fail to be real (or the Hamiltonian wouldn't be Hermitian).
In other words, you know that the Klein-Gordon equation may be derived from the free Dirac equation if you apply the Dirac operator twice. That's why the squared mass in the Klein-Gordon equation has to be the square of a real number: it can't be negative.
So there are various reasons why I think that the claim will go away. They will review the accuracy with which they may determine the timing and the signal will no longer be six sigma (but I can't even tell you whether the wrong timing affects the decays at CERN or the detection in Gran Sasso). Superluminal particles are the ultimate extraordinary claim, it seems to contradict many other observations that have eliminated such a possibility, so some truly extraordinary evidence is surely necessary.
Statistically speaking, six standard deviations is a strong signal but it is still "of order one". For such a new experiment that is measuring something totally unusual, you must always wonder why the signal is just 6 times larger than their error margin. Why are they comparable? Why isn't the accuracy 50 times better than needed? Times that may be measured by clocks in Italy span many possible orders of magnitude. From this viewpoint, "6 sigma" is still too close to "1 sigma" and the big claim is likely to be due to an underestimate of some systematic errors.
Let me say it in other words. When there are claims about deviations from the currently valid theory that violate its basic properties, we shouldn't just wait for some statistically significant result that says "something unusual is going on". We should wait for a big enough accuracy so that the experimenters can tell us what the deviation is and how it phenomenologically depends on certain quantities. For example, they should first say that the speed is \(v=c+a/E_\nu\) and the coefficient \(a\) should be measured accurately enough and by various places and perhaps by independent methods.
There are no tachyons in Nature. Well, to be more accurate, the Higgs boson would be a tachyon if we could spend our lives near the maximum of the Higgs potential, \(H=0\). However, this point is unstable so the Higgs would quickly roll down – and did roll down – to a value where the energy density is lower, to a true minimum. That's why the electroweak symmetry is spontaneously broken – assuming that God gives us His particle. And when we expand around this point, there are no tachyons left.
So what Opera has produced is just a Phantom of the Opera:
Phantom of the Opera, by Mr Martin Chodúr, the 21-year-old recent winner of the Czech and Slovak American Idol, and the 13-year-old recent winner of the Czech and Slovak Talentmania, Miss Patricia Janečková. I assure dear Polish visitors that our beloved shared star Ewa Farna can also sing opera but Ewakuacja may still win. ;-)
Just to be sure, a phantom is a ghost, usually of a deceased person, and – as the singers suggest – it exists inside your mind only.
If you're interested in physics, and physics is the hottest topic in Poland since the times of Maria Salomea Skłodowska :-), try other articles on this blog. For example, my ex-colleague and one of world's top physicists Lisa Randall of Harvard (extra dimensions etc.) guest blogged over here today (Tuesday) about her new book, Knocking on Heaven's Door.
Neutrinos are being mentioned or discussed in 125 articles on this blog.
Update: On Thursday, AP, BBC (who renamed Opera to Orion: fixed) and Reuters (and all other media) finally leaked the news: the neutrinos are claimed to have arrived 60 nanoseconds before the light (18 meters over c). Because this is said to be a 6-sigma signal, their total error margin of the timing should be 10 nanoseconds (3 meters over c); recall that the distance is 732 km. I leave it to the reader to decide whether this accuracy is plausible given the messy birth and detection of the particles. One nanosecond is the duration of one cycle of your iPhone microprocessor, among other things. Ten nanoseconds is 40% of the lifetime of the charged pion or 80% of the lifetime of the charged kaon. I can kind of imagine that they're doing something really silly, like imagining that each pion or kaon lives at least for the lifetime and then it dies. But some of them decay immediately; this error could erase most of the 60-nanosecond discrepancy.
Talk on Friday: watch at 4 p.m. Prague Summer Time; Preprint is finally out (click!)Anna has suggested that their GPS-based timing device may have neglected that the electromagnetic waves are moving slower than c through the atmosphere: if the collaboration did an error in this subtlety, they get an error of exactly the same magnitude to explain the "signal". The index of refraction of the air is 1.0003, so light needs to penetrate a 10-km layer of the atmosphere as it would need to get through 10.003 km of the vacuum which would exactly produce the 3-meter delay. Make the atmosphere a bit thicker because the satellites are not right above your head; add the delays from both directions and you may already produce those 18 meters of error (or most of it).
See #MundaneNeutrinoExplanations for some ongoing Twitter fun such as:
Bartender says: "We don't serve neutrinos here." Neutrino enters a Gran Sasso tavern. :-)
Text below was originally posted on September 19th
Experimental claims on neutrinos that are faster than lightTommaso Dorigo is spreading a six-sigma (6.1, to be more precise) rumor involving neutrinos and his countrymates in an experiment melodically named Opera (well, the result is actually mostly due to French intruders from Lyon, but that's a detail):
A Six-Sigma Signal Of Superluminal Neutrinos From Opera! (Dorigo)The first link points to a Yahoo cache because Dorigo had to scrap his article: as an employee of IFNF in Italy that co-funds Opera, this is a top-secret classified material for him. On Friday afternoon, I returned the original Dorigo link: it works again.
Can neutrinos be superluminal? Ask OPERA (Gibbs)
Supernovas and neutrinos (Strassler)
When we visited Milan, a pigeon managed to deposit some products of its metabolism directly into the pocket of my shorts. This procedure was apparently achieved from a nearly impossible (vertical?) angle – much like if your world line in the spacetime were spacelike instead of timelike.
This experience taught me to be careful about angles measured in Italian towns, especially the towns that are famous for an opera. That lesson will turn out to be useful momentarily – in the case of the spacetime surrounding the Gran Sasso mountain.
A theater that is prettier than La Scala.
And by the way, we were so disappointed by the exteriors of the famous La Scala, the local opera house, that I decided to post a picture of our local J. K. Tyl's Theater in Pilsen which all Pilsner tourists preferred over La Scala. ;-)
Make no mistake about it, this trash talk is a preparation for the soccer match, FC Viktoria Pilsen vs AC Milan, in the UEFA Champions League. We will show them the door much like Napoli did a few days ago ;-) before we will do the same thing with Barcelona, a Catalan village where they thought they could play soccer before they met any players from Pilsen. :-) Unfortunately, BATE Borisov (Belarus) played a tie against Pilsen in the first match of our group; Milan tied with Barcelona in our group, too.
Back to neutrinos
Fine. What Dorigo says is simple. They measured how much time it takes for the tau neutrinos to cause a reaction in Gran Sasso tavern in central Italy. The (currently) tau neutrino \(\nu_\tau\) came together with a mostly muon-neutrino beam originally created by decays of pions \(\pi\) and kaons \(K\) into muons \(\mu\) and muon neutrinos \(\nu_\mu\) at CERN, more precisely in SpS which is a pre-accelerator of the LHC.
They tried to be careful about the timing, measured the delay, calculated the speed. And their resulting speed of the neutrinos is
\[ v = c + 6 \sigma. \] It is greater than the speed of light – by an excess of speed that is six times larger than their total error margin. If this claim were true, the probability that this excess happens by chance is lower than one part in half a billion. Much like the pigeon's feces near the opera house in Milan, the Opera collaboration claims that the neutrinos are moving along impossible angles in the spacetime.
The Opera experiment
Of course, the claim is false although I can't tell you exactly why because I wasn't told any further details about the rumor, either. They have surely screwed the calculation of some timing in the multi-step process.
The claims about a superluminal neutrino have a long history. All those claims in the past seem to be impossible although people are usually unaware of the new evidence that identified the errors in the superluminal claims.
Supernova observations are the most reliable source to tell us that neutrinos are moving slower than the speed of light – because their mass is actually positive. For high energy neutrinos, the deviation of the speed from the speed of light is smaller and the neutrinos arrive simultaneously with the light produced during the supernova explosion. But high-energy neutrinos that are faster than light seem to be incompatible with some observations of the supernovae.
You have to assume that neutrinos have a different speed when they move through the Italian rocks. They're perhaps faster in rocks than in the vacuum, Dorigo will tell you. But rocks are almost transparent for neutrinos, according to everything we know about neutrinos today. It's hard to imagine how the rocks could matter. Moreover, local interactions of the rocks with the neutrinos, if they were relevant, would almost certainly slow the neutrinos down instead of speeding them up.
You may see my answer on theoretical physics stack exchange with a list of theoretical papers that discussed superluminal/tachyonic neutrinos...Opera vs violinist Einstein
More importantly, superluminal neutrinos – or any other particles – contradict special relativity. If you could shoot superluminal neutrinos, they would move along spacelike paths in the spacetime and the Lorentz symmetry would allow you to boost the SpS a little bit so that the neutrinos would be moving backwards in time (there is no qualitative difference between spacelike paths going to the future and those going to the past). By boosting the SpS accelerator, you could create neutrinos that would move backwards in time and you could kill your grandfather before he had his first sex with your grandmother, thus rendering your own existence needed for the homicide inconsistent with the result of the homicide.
This shouldn't happen. Your grandfather in the past has the right to protect his record of having sex with your grandmother – much like the Italian prime minister has the right to defend his record that he was able to do eight women in a single night (he couldn't manage more, he says, even though eleven women, one for each dimension of the spacetime in M-theory, were waiting in the line). So neutrinos should better be slower than light, otherwise the pride of Berlusconi who just left a court (in Milan, of course) would be in trouble. And the pride matters more than his job because he's only a prime minister in his spare time, we were told. :-)
CERN and Gran Sasso are 732 km i.e. 2.44 light milliseconds away from each other.
More than ten years ago, around the year 2000 or so (when I was a Rutgers grad student), there were claims that the radioactive decay of the tritium atoms implied that neutrinos should have been superluminal. The statistical significance of the claims was apparently very low. I don't know what exactly happened with that claim but because no one has talked about this revolution in recent years, I guess that it slowly evaporated. This was the first time when people tried to quantitatively analyze the beta-decay of atoms and not just the bare nuclei so I am sure that they did something wrong, or they were just unlucky with the statistics.
Tachyons and spin
In bosonic string theory, the 26-dimensional ancestor of modern (super)string theory that was randomly discovered in the late 1960s, one has a troublesome particle called the tachyon. It is a scalar particle associated with the field \(T(x^\mu)\) whose squared mass is negative:
\[ m_T^2 = -\frac{1}{\alpha'} = -\frac{T_{\rm string\,\,tension}}{2\pi} \] for the open string tachyon and 4 times larger for the closed string tachyon. At the level of classical particles, the negative squared mass looks like faster-than-light propagation of particles that allows you to inconsistently change your grandfather's bedroom record.
At the level of field theory, a negative squared mass means that the potential (density of potential energy to be more accurate),
\[ V(T) = \frac{m_T^2}{2} T^2, \] acquires a maximum rather than minimum at \(T=0\). That's very bad because this value of the tachyon field becomes unstable. The field wants to roll down and find a true minimum. If it doesn't find a true minimum, it will continue to roll indefinitely. Quantum mechanically, this increasingly fast rolling on the slope looks like a spontaneous creation of tachyonic particles or their pairs with both positive and negative energies.
Superstring theory eliminates tachyons and guarantees that the theory contains fermions as well. But we have never seen a string theory with a tachyon which is not a scalar particle. Is it a coincidence? Can't we have a tachyon with spin? A Dirac tachyon? Or a vector tachyon?
The answer is that it is no coincidence. In string theory, tachyons have to be scalar particles. The reason is that to give the tachyon a nonzero spin, you must excite it by some stringy oscillators that are connected with large spacetime coordinates. But these oscillations raise the squared mass of the resulting particle and because of that, the particle's squared mass can no longer be negative. In bosonic string theory, you may only obtain spin by \(\alpha_{-n}^\mu\) which increases \(N\) by one (minimal amount for \(n=1\)); in supersymmetric string theory, one may also have \(\psi_{-n}^\mu\) where \(n\) may be as low as \(n=-1/2\) in the antiperiodic (NS) sector. However, the ground state energy is less tachyonic in the RNS superstring case and this is enough to bring you to the massless level with just a single excitation, too.
Of course, those calculations only apply to the free string theory – without any interactions. (But the result is still true even in you consider arbitrary orbifolds etc., even in the twisted sectors where the ground state energy is less tachyonic or not tachyonic at all.) Can you have Dirac particles when you turn on interactions? Can you have spin-1/2 fermions which are tachyonic?
Effective field theory vs spinning tachyons
Well, the answer is that this is impossible even at the level of effective field theory. Recall that the kinetic term for a Dirac particle is
\[ {\mathcal L}_{\rm kin} = \bar \psi \gamma^\mu \partial_\mu \psi \] and is independent of the mass. The mass term
\[ {\mathcal L}_{\rm mass} = -m \bar\psi \psi \] has a coefficient of \(m\). Note that the mass is not squared in this case (unlike the case of the scalar tachyonic particle). Consequently, you can't make the world lines spacelike because that would mean an imaginary value of \(m\) and the action would fail to be real (or the Hamiltonian wouldn't be Hermitian).
In other words, you know that the Klein-Gordon equation may be derived from the free Dirac equation if you apply the Dirac operator twice. That's why the squared mass in the Klein-Gordon equation has to be the square of a real number: it can't be negative.
See my somewhat technical report about neutrino oscillations and Majorana vs Dirac character of the neutrinos (nu.pdf) that I presented in the late 1990s.Similar field-theory arguments may be used to show that gauge bosons associated with spontaneously broken gauge symmetries can't be tachyonic, either: in \(m_A^2 = g^2 v^2\), both factors on the right hand side (the squared coupling and the squared vev) are positive. One may probably generalize these arguments to arbitrary spins: the only allowed spin of a tachyonic particle in a Lorentz-invariant theory is zero. Such scalar tachyonic particles make the theory unstable.
So there are various reasons why I think that the claim will go away. They will review the accuracy with which they may determine the timing and the signal will no longer be six sigma (but I can't even tell you whether the wrong timing affects the decays at CERN or the detection in Gran Sasso). Superluminal particles are the ultimate extraordinary claim, it seems to contradict many other observations that have eliminated such a possibility, so some truly extraordinary evidence is surely necessary.
Statistically speaking, six standard deviations is a strong signal but it is still "of order one". For such a new experiment that is measuring something totally unusual, you must always wonder why the signal is just 6 times larger than their error margin. Why are they comparable? Why isn't the accuracy 50 times better than needed? Times that may be measured by clocks in Italy span many possible orders of magnitude. From this viewpoint, "6 sigma" is still too close to "1 sigma" and the big claim is likely to be due to an underestimate of some systematic errors.
Let me say it in other words. When there are claims about deviations from the currently valid theory that violate its basic properties, we shouldn't just wait for some statistically significant result that says "something unusual is going on". We should wait for a big enough accuracy so that the experimenters can tell us what the deviation is and how it phenomenologically depends on certain quantities. For example, they should first say that the speed is \(v=c+a/E_\nu\) and the coefficient \(a\) should be measured accurately enough and by various places and perhaps by independent methods.
There are no tachyons in Nature. Well, to be more accurate, the Higgs boson would be a tachyon if we could spend our lives near the maximum of the Higgs potential, \(H=0\). However, this point is unstable so the Higgs would quickly roll down – and did roll down – to a value where the energy density is lower, to a true minimum. That's why the electroweak symmetry is spontaneously broken – assuming that God gives us His particle. And when we expand around this point, there are no tachyons left.
So what Opera has produced is just a Phantom of the Opera:
Phantom of the Opera, by Mr Martin Chodúr, the 21-year-old recent winner of the Czech and Slovak American Idol, and the 13-year-old recent winner of the Czech and Slovak Talentmania, Miss Patricia Janečková. I assure dear Polish visitors that our beloved shared star Ewa Farna can also sing opera but Ewakuacja may still win. ;-)
Just to be sure, a phantom is a ghost, usually of a deceased person, and – as the singers suggest – it exists inside your mind only.
If you're interested in physics, and physics is the hottest topic in Poland since the times of Maria Salomea Skłodowska :-), try other articles on this blog. For example, my ex-colleague and one of world's top physicists Lisa Randall of Harvard (extra dimensions etc.) guest blogged over here today (Tuesday) about her new book, Knocking on Heaven's Door.
Neutrinos are being mentioned or discussed in 125 articles on this blog.
Italian out-of-tune superluminal neutrino opera
Reviewed by MCH
on
September 22, 2011
Rating:
Reviewed by MCH
on
September 22, 2011
Rating:
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