108. How stringy is QCD string?
HEP-TH/PH — By Dmitry Podolsky on November 23, 2008 at 8:17 pm
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I am in Munich now, so please forgive me for being a bit quiet 
As I’ve explained in this small introduction into criteria of confinement, breaking of the chromoelectric tube (string) connecting heavy quark and antiquark happens through the production of a pair of light quark and antiquark in the strong chromoelectric field. The physics behind this is the following. Suppose we are trying to pull the heavy quark and antiquark from each other. As was explained in this post, the energy of the string grows (almost) linearly as
,
where 
is the string tension. The latter can be estimated as
(1);
it remains practically constant for sufficiently large 
, see for example this Fig., from Bali, 1993 (sorry that the paper is so old, but I like that he has put scales on the plot):
(Note that I want to write the energy density of the chromoelectric field per unit volume as 
instead of 
and neglect the trace in what follows. As you have surely understood at this point, 
is the chromoelectric field strength. The scale 
defines the width of the cromoelectric tube.)
Chromoelectric field in the tube is rather strong (from (1) we find that
(2)
or, if we want to take all the numbers in (1) literally, 
, although it does not have very much sense to keep factors like 
in our estimations), and quark-antiquark pair may be copiously produced in this field inside the tube.
Let us estimate the rate of the corresponding particle production. We can use the Schwinger formula for the probability of particle creation per unit time per unit 3-volume:
, (3)
where 
is the component of momentum of created particles perpendicular to the tube. Since the tube is rather thin (it has thickness of the order 
), this momentum is large – according to the Heisenberg uncertainty relation we have:
.
Since the coupling also contributes to the rate formula, we need an estimation for the coupling as well. Using the Gauss law, we find
. (4)
Then, as follows from (2), the effective coupling 
in the tube. Therefore,
. (5)
Let us first consider the case of heavy quarks 
. Then, only the first term is important in the series:
As I have explained above, this is probability per unit 3-volume and unit time. The probability per unit time is
where 
is the length of the chromoelectric tube that we want to estimate.
The characteristic time between two consequent acts of particle creation is the quantity which is inverse to 
:
If the created quarks were able to propagate to the distance 
during this time, the chromoelectric field of the original tube gets screened by them, and the tube breaks down into two smaller ones connecting the quark (antiquark) of the original pair and the antiquark (quark) of the newly created pair. We therefore find
,
i.e., if quarks are much heavier than the non-perturbative scale 
, then the characteristic length of the tube is exponentially larger than its width . In other words, the chromoelectric tube indeed looks like a string in this limit.
Let us now turn to the case of light quarks 
. In this case, all
terms are important in the expansion (5). If fact, the series (5) can be summated exactly (the sum is basically the polylog function), and in the limit 
we find
(6)
Exercise. Check it explicitly.
Exercise. Find leading order 
correction to (6).
In other words, the chromoelectric tube looks very thin (its length is much larger than its thickness) also in the limit of light quarks.
Let me finally list some other scales I have used during this discussion, since we will probably need them further:
- non-perturbative scale of quantum chromodynamics
(with quarks) is of the order 200-300 MeV; by definition, this is the scale where the effective coupling 
diverges - the interquark potential behaviour becomes linear at scales of the order 1/4 fm; 1 fm is approximately the size of the proton, about 5 inverse GeV
- inverse QCD string tension
is of the order 300-400 MeV, i.e., of the same order as , - chromoelectric field in the string can be easily caculated from the latter: we have
, - what happens in real QCD – should we take the limit of light or heavy quarks? u- and d-quarks both have masses less than 10 MeV, so naively the limit of light quarks holds. However, at distance scales larger than 0.1 fm we need to take the mass of the gluon cloud into account, and the renormalized masses of u? and d-quarks turn out to be of the order 350 MeV or so. So, it look like the heavy quark limit is more relevant in the real life (or, actually, the case
.) Anyway, as I shown, the QCD chromoelectric tube is thin in this limit.
As we see, the only interesting scale of QCD at strong coupling is , and the essence of the mass gap problem I have mentioned in the introduction is to derive the scale theretically starting from the QCD largrangian.
8 Comments
It is not only that QCD is stringy; it is that Nature’s QCD can be aligned to be super-stringy (0710.1526) by using the arrangement suggested in hep-ph/0512065. And probably it is the only number of flavours allowing such arrangement.
Hi Alejandro
I have read your papers, and to say honestly, I do not understand a single little piece in them except the numerology part. Where is SUSY (if it is hidden in the spectrum, what is the scale of its breaking, where are squarks anyway?) hidden in your considerations for example?
) – g\sim 1 in the string, and this is only possible if
As for the number of flavours, the only place the number o flavours enters the considerations above is the fact that I need asymptotic freedom (and IR slavery
N_F < 11/2 N_c
Cheers
Sorry I usually mud my own papers with numerology; in the case here, section 2 of 0710.1526 was introduced to ask people to think if the coincidence between the mass of the muon and the pion could be something with a deeper meaning. In introduces one half of
the point of section 3: that there are the same number, charge by charge, of leptons in the standard model than mesons in the quark model. It is an amusing coincidence. The coincidence is enhanced by looking at the quark sector, where it happens that the number of possible quarks again coincides with the number of quarks in the standard model. So what I say is that there is a possibility to arrange SUSY Lagrangians by putting diquarks and mesons (the QCD strings) in the same footing that quarks and leptons. For instance, any particle having equal interaction strength with spin 1/2 quarks and scalar diquarks would automagically, in the standard model, win the same kind of protection against perturbative renormalisation that the Higgs gets in the SUSY SM.
In this idea, the squarks and sleptons are the lowest energy states of the QCD string; the number of different QCD strings depends on the number of flavours. The arrangement where there is equal number of fermions (the SM ones) and bosons (the QCD strings) only happens for three generations of the standard model and a very massive top quark (so that stable QCD strings terminating in top quarks do not exist). It is a peculiar fact.
Hi Alejandro
Thanks for the explanation.
That’s the key idea of your construction and that is what I do not quite understand (or don’t understand how to realize it). What do you mean by putting them in the same footing – considering composite and fundamental particles at the level of QCD largangian simultaneously? The first ones (mesons) are non-local degrees of freedom (strings) with characteristic width >> inverse Lambda_QCD. Are you saying that one is able to introduce another string connecting a meson and a quark, and the charatersitic length of this string will be larger than the one of the meson?
Cheers
You are pointing right to the problem: It is a half baked idea as it stands. Up to now, my only real point is that there is a coincidence one could try to exploit.
Of course while mesons are color neutral, so they can not be the terminations of a QCD string. On the other hand, diquarks are coloured, and a QCD string should be able to terminate in a diquark. This brings out the worst problem of this idea: what to do about the +4/3 diquarks. I suspect that some part of the -unknown- formalism whould truncate they away, because they are the only ones I can not organize their superpartners into Dirac fermions. For -2/3 diquarks, there are (for each color) 6 of them, and 6 antidiquarks, so I can organise 3 generations. For +1/3 diquarks, it is the same. But for +4/3 there are only 3, plus 3 antidiquarks, so the partner fermion should be 3 generations of a chiral object, which in turn could not exist at all (truncated away), or in anycase it could not be a termination of the QCD string because QCD is not a chiral theory. Arguably indeed.
Well, to resume the answer to your last question: the lagrangian of such “QCD superstring theory” should include the mesons and scalar diquarks, yes, but only the later ones would participate in the QCD interaction.
A further connection could appear in the mechanism to put mass. Such “lagrangian” should have two different ways for mass to enter into the scene. On one side, mass terms could be given to each particle using a Higgs. It implies that the Higgs object of this theory should have an interaction both with elementary and with composite; this is the thing I was first thinking when I told about “same footing”. On the other hand, each pair of particles, meson, diquark or even barion (diquark-quark), gets mass from QCD in the usual way. It should exist some tuning (inside the Higgs mechanism, probably) forcing that the mass got via the later does coincide with the mass got via the former mechanism.
When, two years ago, I dedicated some active thinking to this idea, I really was driven to deep into the traditional superstring as the ideal formalism for such “lagrangian”. A lateral point here was that at all I am allowing for 5 different labels for string terminations: u,d,s,c,b. And a paper of Marcus and Sagnotti centuries ago pointed out that the quantisation of an string with such n different labels automagically implied a SO(2^n) symmetry. Thus the touted SO(32) symmetry of the superstring would be simply the stringy version of flavour. But again, this path drove me to nowhere, at that time.
Hi Alejandro
Indeed so.
Now, suppose I consider a theory with Meissner effect (superconductivity) instead of the theory featuring dual Meissner effect (QCD). In QCD you want to consider fundamental degrees of freedom (quarks) and composite ones (diquarks) on the same footing in the QCD Lagrangian. I have an impression that one simply cannot consider Cooper pairs and electrons simultaneously – after Bogolyubov transformation diagonalizing fermionic Hamiltonian with phonon interaction taken into account only Cooper pairs survive.
Bogolyubov transformation shows what is the correct vacuum of the theory. At T>T_c Cooper pairs dissolve, and correct vacuum is the trivial one (degrees of freedom are electrons and holes), while at T<T_c the vacuum is superconducting, and correct degrees of freedom are Cooper pairs.
The question is why one could consider quarks and diquarks simultaneously in QCD?
Cheers
A thinking… perhaps it is not about having quarks and diquarks simultaneusly, but about having the same number of degrees of freedom in both sides of the transition, either T greater or smaller than T_c.
I have been glancing randomly some papers of Shifman and related teams. It seems that there is much work in diquarks, but nobody worried about this specific situation. Not rare, because there is anyway the question of the use of a mass criteria to discard some pairs (and particularly the need of discarding the diquarks uu uc ut cc ct tt without discarding the equivalent dd ds… set).
Hi Alejandro
Sorry for the slow response. Could you give me a couple of nice references about diquarks to read?
Cheers
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