The Virtue of Being Virtual

The photon that left your computer screen and just hit your retina now certainly was a ‘real’ photon. Right? Well, not so fast. Sketch the Feynman diagram for the process. The photon was emitted by an atom in the screen, and absorbed by an atom in your retina. The diagram is indistinguishable from that for a virtual process- it’s all a matter of scale.

This is clearly explained in Theory of Fundamental Processes by Richard Feynman (see Chapter 20, “Virtual and Real Photons”, there are also similar remarks in Chapter 3 of the master’s Feynman Lectures on Gravitation ; if you are aware of a similar more modern discussion I’d appreciate a reference in the Comments). The photon in question had a frequency of around 1015 s-1. It traveled the 30 cm from your screen to your eye in about 1 ns, which means that it completed about 106 cycles. That means, modulo factors of 2π, that its relative spread in energy ΔE/E ~ 10-6, i.e., to within about a ppm it is a ‘massless’ real photon. But this tells us that any detected photon is only approximately ‘real’- we could make the photons emitted by your screen and detected by your eyes a factor of 10 or even 100 more ‘virtual’ simply by placing our noses (well, eyes) on the screen.

This close connection between real and virtual photons has been exploited in a recent publication by the PHENIX Collaboration

The observation of direct photons has long been held as a sine qua non of QGP formation. For instance, in QCD Made Simple from the August 2000 issue of Physics Today, Frank Wilczek writes

“Using fundamental aspects of QCD theory, similar to those I discussed in connection with jets, one can make quantitative predictions for the emission of various kinds of “hard” radiation from a quark­-gluon plasma. We will not have done justice to the concept of a weakly interacting plasma of quarks and gluons until some of these predictions are confirmed by experiment. “

While it is not completely clear what ‘hard radiation’ Wilczek is referring to, it’s safe to assume that direct photons would be a leading candidate. Of course we now know that sQGP formed at RHIC is anything but a ‘weakly interaction plasma of quarks and gluons’, which makes the measurement of direct photons from it all the more compelling.

However, the detection of direct photons from the plasma is notoriously and famously difficult. (The article Direct Photon Production in Relativistic Heavy-Ion Collisions provides a review of direct photon with a nice discussion of the experimental issues involved.) Photons from a plasma at temperature T have a mean energy ~3T . For plasmas with temperatures of 200-400 MeV, this results in the photons of interest being swamped by the much more abundant decay photons from π0 → γγ and η → γγ . It is impossible to know on a photon-by-photon basis which is a decay photon and which is direct, so experimentalists are forced to carefully measure the background from π0‘s, η’s and other sources, subtract them, and hope that the systematic error on the residue does not encompass zero. With a sufficiently segmented detector and a sufficiently thorough analysis, this is possible in Au+Au collisions for transverse momenta above ~2 GeV/c, which alas is not the regime of thermal production. (All is not lost- far from it! The observation that high transverse momentum direct photons are produced at a rate consistent with pQCD is one of the touchstones for the entire “jet quenching” enterprise at RHIC and soon the LHC.) But even in the window of opportunity ~1 < pT <~3 GeV/c where it is expected that thermal spectrum has optimal strength relative to the falling decay background and the emerging pQCD photons, it is difficult to obtain unambiguous evidence of thermally produced photons.

The virtue of a measurement of virtual photons is that it presents one more knob to turn in optimizing the signal to background: the mass me+e of an e+e pair, which is just that of its parent virtual photon γ*e+e. This is by no means obvious- one’s first thought would be that there would be just as much, if not more, background in this sector from the “external” conversions in the detector material π0 → γγ→ γe+e , along with the unavoidable “internal conversion” or Dalitz decays of neutral pions π0 → γe+e . The trick lies in making this a feature rather than a bug- such pairs necessarily must have masses less than that of the parent π0, so there is a region of opportunity, roughly mπ0 < me+e < 2mπ0 where the background is minimal. Studies by PHENIX show that this is indeed the case- the background from a “cocktail” mixed from measured sources (mostly Dalitz decays of π0‘s and η’s) drops precipitously for me+e > mπ0 . In p+p collisions, the yield of pairs above the pion mass is consistent with the predictions of NLO pQCD, while in Au+Au collisions there is a clear excess for pairs with transverse momenta below ~3GeV/c.

Translation of these measurements for excess e+e pairs to a statement for photons requires a) an assumption, namely that all such pairs result from internal conversion of virtual photons and b) a prescription, for converting the pair yield into the corresponding spectrum for real photons. The assumption is plausible, since there are no other known sources in the mass range ~mπ0 < me+e < ~2mπ0 studied by PHENIX. The prescription is straightforward, as there is a standard relation between the spectrum of real and virtual photons from a source,

$latex { dR \over d^4q} = {2\alpha \over 3\pi}\,{1 \over M^2}\,(1+{2m^2 \over M^2})\sqrt{1-\frac{4m^2}{M^2}}\,\left({dR \over d^3k/\omega}\right)&s=2$

where the rate of the real photon production is the last factor on the RHS, and the virtual photon with four-momentum q is detected as leptons of mass m with invariant pair-mass M. (This is clearly discussed in Chiral Symmetry Restoration and Dileptons in Relativistic Heavy Ion Collisions (see Section 4.3); the place I always return to for reference is Photons and lepton pairs: The Deep probes of quark – gluon plasma by Vesa Ruuskanen, see in particular Eqs. 14 and 15 in this scanned version. Again, other references would be appreciated as comments.)

By following this procedure, the PHENIX collaboration infers the direct photon signal in the region 1 < pT < 5 GeV/c for both p+p and Au+Au collisions. The direct photon yield for p+p collisions is consistent with the predictions of NLO pQCD, which provides some support for the validity of the method. In Au+Au collisions the clear excess of pairs translates into a “thermal” (exponential in pT) signal corresponding to a (time-averaged) temperature T ~ 220 MeV, consistent with hydrodynamic models starting with Tinit ~300-600 MeV. So if you’ve managed to stay with this somewhat complicated discussion, it would appear that this nice technique has provided evidence for thermal photons from the sQGP.

(While the focus is on the excess above the NLO pQCD yields in Au+Au collisions, it is amusing to note that i. the pQCD calculation seems to work down to the lowest measured pT‘s, a quite surprising result and ii. these are the lowest measured transverse momenta for direct photon yields in p+p collisions, which is a ‘direct’ result of the novel technique employed.)

Finally, I can’t resist returning to virtue of being virtual. Of course this entire “journal” is virtual, the reader will make his or her own judgment as to its virtue. But the point I’d like to make is that our physics descriptors of “virtual” and “real” photons is very nearly a precise inversion (at least in English) of the adjectives real and virtual. What could be more virtual than a photon that is never detected? But that is the only photon that has a chance of being truly massless. What could be more real than a photon that you have detected? But that photon is necessarily virtual, as noted above. Left as an exercise to the reader is to estimate how many cycles the γ*‘s studied by a PHENIX complete before “internally converting”.

1. September 4th, 2008 | 11:22 am

Dear Bill,

I read your article “The virtue of being virtual” with great interest. I appreciated, in particular, your reference to Feynman and the associated play on words. However, your beautiful concluding sentences, again reduced to the spirit of the Uncertainty Relation, remained ambivalent, without really addressing the duality in our approaches towards thermal radiation in a more concrete way.

The fireball formed in nuclear collisions is full of photons, real and virtual. Once they leave, however, a clear selection has taken place: they are either essentially real OR virtual, the latter appearing as pairs of oppositely charged point-like leptons (I discard strongly interacting charged particles). In vacuum, the mixture of the two in a given event class is invariant and does not really depend on how close we “place our noses” (alias detectors) to the fireball. “The virtue of being virtual” then transforms into “The virtue of being lepton pairs”, and this is the more precise nomenclature which I am now going to fill with some additional substance. I will abstain from a discussion of hadron-decay background.

The relative virtues of dileptons and real photons fall into two categories – dynamics and kinematics. Both tend to make the physics of dileptons more rigorous and more rich than that of real photons.

• Dynamics

The lowest-order rates for the production of lepton pairs are ∝αem2. Prime examples: Drell-Yan early on; qqbar-annihilation in the plasma; pipi annihilation in the hadron phase. Vector mesons are filtered out via their 1– – quantum numbers. The cleanness of the examples is equivalent to the cleanness of the respective time reversed reactions, using electron-positron collisions. Of course, there are higher-order processes.

The lowest-order QCD rates for direct photons are ∝ αem x αs. Prime examples: QCD Compton early on; related graphs in the plasma. It goes without saying that the sole existence of αs induces major uncertainties. Photon rates in the hadron phase are hardly less uncertain.

• Kinematics

Dileptons are characterized by two independent variables, M and pT. M can be rich in resonance structure, pT never. Prime examples: J/ψ early on; rho in the hadron phase, the best probe of chiral symmetry restoration. Just as important: correlations between M and pT for continuum pairs, on a statistical basis. Prime examples: mass-dependent radial flow (or its absence), a handle on the emission region of lepton pairs including the issue of partons vs. hadrons; mT-scaling for thermal radiation in contrast to strict factorization between M and pT in Drell-Yan (applicable for M > 1 GeV).

Real photons are characterized by just one variable, pT. Evidence for thermal radiation can solely be based on theoretical “cocktails” of various contributions, including hard pQCD processes. There is no characteristic flag for any source except for its shape in pT space.

Internal conversion of photons into lepton pairs is a hybrid in this environment. Dynamically, as directly reflected by the wording, it belongs into the (direct) real photon category, but with a lowest-order QCD rate now ∝ αem2 x αs. However, the “cocktail” of other direct photon sources has here to be supplemented by the dilepton sector where internal conversion is far from being the only direct source, irrespective of the mass window. Kinematically, conversion pairs share the property of two variables, M and pT, with all other dileptons, and could therefore supply more insight than the mother real photons alone, even if the masses due to pT(Photon) > M are small.

I hope that these remarks may help to embed internal photon conversion, the method propagated in your article, into a wider perspective.

Best regards,
Hans