This blog and other sources occasionally mention excesses, possible (or real?) deviations of the observed quantities away from the theoretical predictions. They're extremely rare and we expect most of them to go away when more data is collected, anyway. But the idea that some of of them are real is surely what keeps many people going.

However, in most of these discussions, one subtlety is neglected: deviations from predicted values may have two signs. Plus and minus. So in principle, deviations from the Standard Model may be either excesses (surpluses), or deficits (shortages) of events.

I actually feel that the possible deficits are generally underestimated, understudied, and perhaps completely overlooked as hypothetical routes by which Nature may show that She's more complicated and subtle than Her currently valid caricature.

Why do people assume that deviations should be excesses? Well, it's simple. If you calculate the probability (expressed as the cross section \(\sigma\)) that the proton-proton collisions result in a certain final state, you're summing over the intermediate histories. Some of the histories resemble the Standard Model.

But if there are new particles, they may appear in the intermediate states which contribute some new possible histories that affect the total cross section. So we tend to think that\[

\sigma_\text{total} = \sigma_{\rm SM} + \sigma_\text{new physics}.

\] So you see that the total cross section is greater than the cross section calculated from the Standard Model; the total probability is greater because of the extra probability that the new particles appear as intermediate particles.

Well, the only problem is that the formula above is wrong.

It would be right in a world governed by classical physics but the world is quantum, stupid. In quantum mechanics, the calculation of the probabilities follows a slightly different, fancier but equally (if not more) rigid procedure:\[

\eq{

\sigma_{\rm SM}&= \left|{\mathcal M}_{\rm SM}\right|^2\\

\sigma_{\rm total}&= \left|{\mathcal M}_{\rm SM}+{\mathcal M}_\text{new physics}\right|^2

}

\] One must first calculate the complex probability amplitudes for the Standard Model and for the new physics, add them, and then the result of this addition must be squared. The absolute value is the total cross section.

This quantum formula is a straightforward generalization of the wisdom you should know from the double slit experiment. Feynman liked to say that all the wisdom and subtleties of quantum mechanics may be extracted if you carefully think about the double slit experiment and this insight is surely no exception.

Because the complex amplitudes \({\mathcal M}\) must be first summed, there is some room for negative interference. If the Standard Model and new physicsi amplitudes have the opposite sign, their effects will tend to cancel or partially cancel and the total cross section will be smaller than the cross section in the Standard Model.

One should understand that the amplitudes \({\mathcal M}\) depend on the energy and momenta of the particles in the final state and the dependence is usually different for the Standard Model contribution and the new physics contribution. If the relative phase between these two contributions oscillates quickly and chaotically, we will effectively return to the classical \(\sigma+\sigma\) formula for the cross section. That's another reason why the excesses are the "norm" even if you allow "some logic" that also tolerates deficits.

However, there exist very good theories with reasons why you should expect really visible deficits, too. String theory has been the most famous example for almost 45 years. In fact, these deficits were the main reasons why string theory was abandoned as a theory of the strong force in the early 1970s. It was too well-behaved. String theory was too sexy for her love, for her shirt, for your party, Milan, New York, and Japan, and for her car. ;-) [Lyrics.]

(I'm a Motl you know what I mean And I do my little turn on the catwalk.)

The technical problem was that string theory has always been good for suppressing the wild (and consequently often divergent) high-energy behavior of quantum field theories. In fact, even for the scattering of hadrons, string theory would be predicting a nicely suppressed cross section of the type \(\sigma\sim\exp(-E/E_0)\) – I don't want to be too accurate because refinements of this formula could easily lead me to the reproduction of a whole introductory string theory textbook – while quantum field theories would prefer power laws like \(\sigma\sim E^{-n}\).

However, it was found out that the quantum field theories' prediction was actually right and string theory's prediction was wrong for the high-energy behavior of the hadrons' cross sections. The decrease in the real world (and in QCD) follows a power law which is why there seems to be "hard seeds" (partons) inside the hadrons which is why QCD, the theory of forces between quarks, was adopted as the winning description of the strong force. The stringy exponential decrease probably kicks in at some point but the point could be close to the Planck scale – inaccessible to any currently imaginable experiments.

(By the way, when the connection between string theory and gauge theories was reborn in the late 1990s, people realized that string theory may also reproduce the power law behavior if one appropriately embeds it in the AdS space and integrates over the location of the interaction along the radial, holographic dimension of the AdS space.)

The exponential decrease of stringy cross sections at high energies may be explained by the interference between different virtual particles (with arbitrarily high spins). Roughly speaking, the sum over \(j\), the spin of the messenger particle, takes the form of the Taylor expansion for \(\exp(-E/E_0)\). The exponentially decreasing behavior may be calculated in many other ways, too.

Now, I don't want to claim that the CMS paper below is any indication of string theory. It just happened that the string theory's softness is my best example for "cross sections smaller than previously thought", i.e. for deficits. Let me finally make the point. Today, CMS released the following paper:

Search for pair-produced vector-like quarks of charge \(1/3\) in lepton+jets ﬁnal state in \(pp\) collisions at \(\sqrt{s} = 8\TeV\)So they're looking for new types of quarks whose charges coincide with the charge of the bottom quark but these quarks are "vector-like". It means that the left-handed and right-handed two-component spinors describing this quark (pieces of the Dirac spinor) actually carry the same charges under the Standard Model so the interaction isn't \(V-A\) (vector minus axial vector, i.e. maximally parity-violating) like it is in the electroweak theory; instead, it is just \(V\) i.e. parity-preserving.

There are various "big picture" theories that motivate the hypothetical existence of such quarks by deeper reasons. For example, such new fermions have to occur in theories based on deconstruction. Nanopoulos et al. also like to predict new vector-like matter.

Mr CMS detector and the experimental physicists whom he employs have looked for signs of such a new quark or, more precisely, signs that such a new quark appeared after the collision and decayed to\[

\eq{

b'&\to tW\\

b'&\to bH\\

b'&\to bZ

}

\] where \(b',b,t,W,H,Z\) are the new vector-like quark, the usual bottom quark, the usual top quark, the W-boson, the newly found Higgs boson, and the Z-boson. Or some combination of these decay channels. Moreover, they focus on subsequent decays of the heavy SM particles that involve electrons or muons. No positive evidence supporting such a new particle has been found (I would bet that no such particle exists in Nature but I am extremely far from being any certain about any such claim).

However, Figure 3 is interesting, anyway:

*Click to zoom in.*

Look at it. You see the Brazil bands around the dashed Standard Model prediction that indicate 1-sigma (68% confidence level) and 2-sigma (95% confidence level) intervals. On the \(x\)-axis, you always see the mass of the hypothetical new quark. The \(y\)-axis shows the upper limit on the cross section of the production of the new quark followed by the appropriate decays.

The black wiggly line is the upper limit as extracted from the actual 2012 CMS data. It should normally fit into the 2-sigma intervals in 95% of cases. However, especially in the upper right and lower diagrams, you see that there are significant – in fact, 3-sigma – deficits relatively to the Standard Model. Such deficits should occur rarely. The pictures show the upper limits on the cross section of production of the new quark assuming that the quark always decays to \(bH\) and \(bZ\), respectively.

If you believe that the deficits are significant and not just downward flukes, the interpretation is the following:

Nature is behaving almost like the Standard Model but it really wants to counteract all possible accusations that it harbors a new vector-like \(b'\) quark that decays to the \(b\) quark and a heavy neutral SM particle (Higgs or Z-boson).If the deficit is a real signal of something, Nature finds it politically incorrect to boast final states that could just resemble a new \(b'\) quark that decays to final states including an ordinary \(b\) quark. The Standard Model isn't good enough for Her to debunk the accusations so She suppresses the processes even more than the Standard Model does.

As hinted at the beginning, the only plausible way to lower the Standard Model prediction is negative interference. There would have to be new processes that look much like the presence of an intermediate \(b'\) quark and their phase should be (almost) opposite to the phase of the Standard Model contribution (which must mimic the process without anything like a \(b'\) quark).

I don't know what the negatively interfering histories could be. But even if the deviation grew, it's possible that there's no new sign of new physics here. It could just mean that the experimenters used some calculation of the Standard Model prediction that forgot about the mixed terms and interference itself.

But things could be hypothetically more interesting, of course. It could be a \(b'\) quark of mass in the \(500\GeV\) through \(600\GeV\) interval where the deficit is seen but one whose amplitude is negatively interfering with the Standard Model which could produce the deficit (although it's hard to believe that the phases could be anticorrelated so nicely for whole intervals).

Don't get me wrong, I do think that this deficit will disappear after more data and/or corrections of the predictions. But I do insist on my thesis that a deficit is a comparably conceivable manifestation of new physics as a surplus that could become the core of the first future experimental shock we are waiting for. If that turned out to be the case, people could say that it is something that almost no one was able to expect. Almost. ;-)

## snail feedback (6) :

"By the way, when the connection between string theory and gauge theories was reborn in the late 1990s, people realized that string theory may also reproduce the power law behavior if one appropriately embeds it in the AdS space and integrates over the location of the interaction along the radial, holographic dimension of the AdS space"

Notwithstanding, spacetime is not really AdS and so ST can be just a good approximation for QCD. Actually, at best of my knowledge ST can get GR + corrections, but (even 45 years later) not QCD + corrections ;-)

Sorry, you're clearly mixing two different applications of string theory where string theory enters differently - the description of gravity (or all forces) in the nearly flat space we know; and the description of physics normally explained by QCD only (or gauge theories) in terms of strings.

In the AdS/CFT sense, string theory gives an equivalent description and the spacetime *is* really AdS (for CFTs; or asymptotically AdS in QCD etc.). This is really the *point* of the AdS/anything correspondence that for a string-based description of non-gravitational systems to be accurate, the spacetime *has to be* AdS and it *is sufficient* when it is AdS.

In fact it is a really beautiful paper, thanks for pointing it! As authors state: "the high-energy behavior in confining gauge/gravity duals is remarkably QCD-like". So, as I have said earlier: it is still not QCD despite of (some) remarkable similarities. On the other hand, this differences become much more explicit (and embarrassing) at quantitative level when computing (for instance) the glueball spectrum from AdS/QCD analogous ... while these results seem QCD, their ordering and absolute magnitudes are not correct. If such compactifications schemes (i.e. AdS/QCD) were rigid enough (instead of hand waving approximations) one could use them to seriously confirm/falsify ST as a theory of hadronic interactions. So, unfortunatelly, even after 45 years of many theoretical development ST produces "almost-QCD" but not real QCD =(

Thanks for this very interesting nice article Lumo :-)

I did not know that deficits can be that cool, so maybe valley hunting is an even more fun sport than bump hunting ha ha ...!

Concerning "String theory was too sexy for her love, for her shirt, for your party, Milan, New York, and Japan, and for her car. ;-)" just note, that I HAVE a T-shirt with a nice stringy scattering amplitude printed on it :-P! But generally you might be right, as at CERN for example they are way too coward to sell T-shirts, caps, mugs, etc with BSM or even stringy stuff printed on it ... :-D

Cheers !

LOL, valley hunting sounds cool. Another disadvantage I didn't mention is that the "valleys" typically don't immediately tell us why they're there, who's the culprit.

Perturbative string theory is OK for shirts but nonperturbative string theory could be too sexy for your shirt. ;-)

Lubos,

Couldn't it be said, apropos the applicability of the double slit experiment, that Schrödinger even extended it into his personal life?

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