I have started to study the AdS/CFT correspondence by reading Minahan's introductory review (arXiv:1012.3983). The first equation of that review is the expression for the leading contribution to the $\beta$-function:
$$
\beta(g)=-{g^3\over16\pi^2}\left({11\over3}N-{1\over6}\sum_iC_i-{1\over3}\sum_j\tilde C_j\right),
$$
where $C_i$ are quadratic Casimirs due to bosons and $\tidle C_j$ are those due to fermions. The author cites Gross and Wilczek (1973) as a source of the formula. But there it looks a bit differently:
$$
\beta(g)=-{g^3\over16\pi^2}\left({11\over3}N-{4\over3}\sum T_j\right),
$$
where $T_j={d(R_j)\over d(G)}\tilde C_j$, $d(R)$ and $d(G)$ are dimensions of the representation and of the group.
Forget about the boson contribution (Gross and Wilczek were not interested in it). But the fermion contribution contains there the factor $-4/3\times{d(R_j)\over d(G)}$ instead of $-1/3$ in Minahan's article. The factor 2 is related to the fact that Gross and Wilczek considered Dirac fermions, while Minahan considers the Weyl fermion. Ok. But we are left with the extra $2{d(R_j)\over d(G)}$ factor, which I am unable to cancel.
Could anybody explain me this factor? May it be related somehow to supersymmetry?
I have seen the quadratic Casimir operators defined with different normalizations. F.i., most physicists would say that C_2 == J_x^2 + J_y^2 + J_z^2 = 2 in the adjoint (spin-1) representation of SO(3) or SU(2). Others normalize it to be 1 in the adjoint representation of any simple Lie group.
Perhaps the discrepancy is related to such different conventions?
No, because in both texts the Casimir is assumed to be equal to N in the adjoint representation of SU(N) (and for this representation C(R) and T(R) coincide).
Then I would trust the formula which predicts asymptotic freedom for QCD with up 16 generations of quarks (Dirac fermions in the fundamental representation).
Surely, it follows from Gross and Wilczek. I am sure, Minahan cites a correct formula as well, but for which theory? No Lagrangian, no details. I guess, he means N=2 (?) supermultiplets rather than individual particles, but there is no explanations in the text.
What you can (we all can) learn from this is to specify the context very explicitly (and preferably very clearly) when presenting results. More likely than not, it is the author who will be the bewildered reader a few weeks down the line... :-)
In a case like this, I wouldn't be shy to ask the author himself, if possible.
Minahan's formula, (for the beta function, not the Bethe function :) ) as he states, is for N=4, super Yang-Mills theory, with SU(N) gauge group, where the fermions (along with all other fields) are in the adjoint representation of the gauge group. For a review cf. http://arxiv.org/abs/hep-th/9912271, where the particle content is discussed, especially, in section 2. N=4 super Yang-Mills theory has only the gauge multiplet, in the ``pure gauge theory'' and that leads to a unique Lagrangian.
Of course the particle content controls the symmetries-in particular the supersymmetry-so one check of the expression is that the contribution of gauge fields, scalars and fermions is such that the beta function vanishes, for N=4 super Yang-Mills.
Stam, thanks for the reference. I'll study it.
By the way, my kind regards to Misha Volkov, Pascal Baseilhac and Oleg Lisovyy.
Dear Misha, glad to have been of any assistance-many greetings and regards to you, also. Another nice reference is, of course, Gross' Les Houches lecture notes of 1975. I think that the extra factor of 2 may come from the remark in d'Hoker and Phong that the contribution of the 4 Weyl fermions is equivalent to that of 2 Dirac fermions-but I'd like to check all factors, of course.
Dear Stam, thanks a lot. I guess I should learn supermultiplets accurately.
Surprised no one corrected the title: It's Beta function, not Bethe function.
Sorry, I just misprinted while typing too fast. For me Bethe equations is a more familiar thing than beta-function. ;-)
You may see the following reference.
The Two Loop beta Function for a G(1) x G(2) Gauge Theory
D.R.T. Jones, Phys. Rev. D25, 581 (1982).
Uhhh... Finally, equation (2.1) in Minahan's review is incorrect. The N=4 supersymmetric theory possesses only 4 chiral fermions (instead of 4 chiral and 4 antichiral, as Minahan claims), as it is written in lectures D'Hoker and Phong and in an old paper by Sohnius and West. This is now consistent with the Gross-Wilczek expression. It would be nice to find, where the boson contribution was correctly calculated, but I'll do it later.
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