Talk:Lattice gauge theories

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    Reviewer A

    Reviewer report on Lattice QCD by P. Majumdar and P. Weisz

    The authors have made an impressive effort in trying to collect the present knowledge and results regarding the numerical simulation of strong interactions on a lattice and in making it accessible to a general public.

    There are of course parts and aspects which can be improved and I will try to make suggestions to the authors in this direction.

    Major suggestions:

    1) The Section on "Confinement" ends claiming that the question about the confinement mechanism, and how to detect it, is still unanswered. While this is true, it may be fair to add that some mechanisms have been proposed, e.g. dual superconductivity or vortex condensation.

    In addition, in the same Section, it may be useful to specify that the center transformation acts only at a fixed time slice.

    2) The Section on "Computer facilities" ends by saying that graphics cards can in principle provide a Teraflop of computing power, "but how much of that can be finally harnessed is yet to be seen". Actually, in the literature one can already find various papers providing precise benchmarks for production codes running on GPUs, both for pure gauge and full QCD, so the status of lattice QCD on GPUs is more concrete than what the authors depict.

    3) In Section "Topology" the authors write a probability distribution function (Eq. 69) for the topological winding number which is apparently gaussian in the infinite volume limit. However it is well known that non-gaussian corrections turn out to be essential in order to get a correct description of the dependence of the QCD free energy on the topological theta parameter, which has a fourth order term, related to non-gaussian corrections, which is non-zero and has been measured consistently in various papers. I suggest the authors to spend a few more words about this in order that the point be well clarified to the reader. In the same Section, it would be fair to mention, apart from the latest results based on GW definitions, also early determinations of the topological susceptibility obtained by means of gluonic operators (e.g. by cooling or by subtracting renormalizations).

    5) In the Section "In Brief" the authors very briefly discuss some topics. Some of them represent quite important research fields for the Lattice community, e.g. "Finite density", to which the authors dedicate just a few lines. I think that it would be unfair and unfeasible to ask the authors to cover every field of Lattice QCD with the due importance and detail. On the other hand I think that it would be fair if the authors stated quite clearly and right from the beginning that they will not cover topics like QCD at finite temperature and density, indicating alternative references for the interested reader.

    Minor suggestions:

    1) In Eq. (5) the argument of the exponential on the right hand side is not dimensionless, maybe a factor "a" is missing.

    2) There seems to be a factor 2 missing in Eq. (6)

    Reviewer B

    The authors have written a very good overview over Lattice Gauge Theories for Scholarpedia, and have covered the central issues of this topic.

    I have just a few suggestions:

    1) The section about Symanzik improvement is rather short. Here some more explanations would be welcome.

    2) In the discusion of the Gell-Mann-Oakes-Renner relation, Eq. 21, it should be mentioned that this relation is not exact but represents the leading term of an expansion in quark masses and their logarithms.

    3) Concerning Eq. 22 and related text, it should be said that d denotes the number of space-time dimensions.

    4) Eq. 66: What is \bar{a} ?

    5) In most cases, references to journal articles are not given, which is ok for a Scholarpedia review. But in some sections, e.g. Algorithms, various special references are given. I would propose to deal with this this more homogeneously.

    And some technical observations:

    6) Some text or equations extend beyond the right margin: Eq. 12, 54, and before Eq. 42, footnote 12.

    7) Figure 2 is not visible in the main article.

    8) In "Computer facilities": the link "here" does not work.

    9) Links to figures do not work at various places.

    10) References to Scholarpedia articles by Faddeev and Wilcek do not have proper links.

    11) Links to articles on arXiv are not ok.


    Reply to Reviewer A

    Thank you very much for your comments. Our responses are enumerated below.

    Major Suggestions

    1) We have added the line :

    Proposed mechanisms for confinement typically require condensation of such gauge configurations in the YM theory ground state and go under names such as "dual superconductivity", "vortex condensation" etc.

    We have accordingly changed

    Consider a transformation $U_4(x,t) \rightarrow zU_4(x,t)$ for all $x$ with $z\in\mathbb Z_N$ the center of $SU(N)$.


    Consider (at a fixed time slice $t=t_0$) a transformation $U_4(x,t_0) \rightarrow zU_4(x,t_0)$ for all $x$ with $z\in\mathbb Z_N$ the center of $SU(N)$.

    2) We have rewritten the lines on GPU computing as :

    A relatively new development is to use the graphics card of a modern PC for computing (Egri, 2007). The Graphics Processing Unit (GPU) brings a peak computing power of one teraFLOPs to one's desktop (Clark, 2010). Many lattice collaborations are now using the GPUs in varying degrees, mainly for the computation intensive part of their programs. A single GPU typically gives a performance of about 100 CPU cores.

    3) We have modified eq.(69) from $$ \left\{1+{\rm O}(V^{-1})\right\} \qquad {\rm to} \qquad \left\{1-\frac{c_4}{8V\chi_t^2}+{\rm O}\left(V^{-2},\frac{\nu^2}{\chi_t^2}\right)\right\}$$

    and added the sentence:

    Here $c_4$ appears in the expansion of the QCD free energy with small topological theta parameter $E(\theta)=\frac12\chi_t\theta^2+\frac{1}{24}c_4\theta^4+\dots$\,, and this has been measured in various simulations.

    Following the suggestion of the referee, after our discussion of measurement of $\chi_t$, we have now added :

    We note that earlier determinations of the topological susceptibility obtained by means of gluonic operators (e.g. by cooling or by subtracting renormalizations) gave results in the same ball-park.

    4) We have added the following lines in our abstract:

    We apologize for the fact that have not covered many important topics such as QCD at finite density and heavy quark effective theory adequately, and mention some of them only in the last section "In Brief". These topics should be considered in further Scholarpedia articles.

    Minor suggestions

    1) We have included a factor $ag_0$ before the integral in eq.(6) and put in $g$ in the definitions of the covariant derivative for consistency.

    2) In eq.(7) we have changed: $\frac{1}{g_0^2}\ldots \to \frac{2}{g_0^2}\ldots$

    Reply to Reviewer B

    Thank you very much for your comments. Our responses are enumerated below.

    1) We cannot think of any way to expand this section without going into a lot more details. We would therefore prefer not to expand this section (unless there is a specific suggestion).

    2) We agree. Accordingly we have changed the sentence before eq.(21) from :

    The squared masses of the bosons are linearly proportional to the quark masses $\ldots$


    The squared masses of the bosons tend to zero linearly in the quark masses;

    and eq.(21) to $$ m_\pi^2 = f_\pi^{-2}(m_u+m_d)\langle\overline{u}u\rangle+{\rm O}\left(m_q^2\ln^{}(m_q^2)\right)\,.$$

    3) We suppose the referee means eq.(32). We have modified the text before this equation from

    $\quad$(Lüscher) term $\quad\to\quad$ (Lüscher) term ($d$ is the number of space-time dimensions)

    4) We have included after eq.(66), the line :$\qquad$ Here $\bar{a}$ is as defined in eq.(28).

    5) We have tried to do this as the referee has suggested and removed the references to the papers restricting ourselves to books and review articles. Only some articles which are new and whose results cannot be found in books and review articles are referred. It is envisaged that as these results find their way into books and articles, the references to these articles can be replaced accordingly.

    6 - 11) We have corrected all the technical points the referee has pointed out.

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