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Pos(LATTICE 2008)057 the Theory Changes Fro Confining to F N 8 and 12 The Conformal Window in SU(3) Yang-Mills PoS(LATTICE 2008)057 Ethan T. Neil∗ Sloane Physics Laboratory, Yale University, New Haven, CT 06511, United States E-mail: [email protected] Thomas Appelquist Sloane Physics Laboratory, Yale University, New Haven, CT 06511, United States E-mail: [email protected] George T. Fleming Sloane Physics Laboratory, Yale University, New Haven, CT 06511, United States E-mail: [email protected] Yang-Mills theory is known to exhibit very different behavior as the number of light fermion flavors Nf is varied. In particular, for sufficiently large Nf the theory changes fro confining to conformal behavior in the infrared. We have obtained results for SU(3) Yang-Mills, based on the study of the running coupling through the Schrödinger Functional approach, that the transition between these two phases occurs between 8 and 12 flavors. I will present the details of our staggered fermion simulations at Nf = 8 and 12. The XXVI International Symposium on Lattice Field Theory July 14 - 19, 2008 Williamsburg, Virginia, USA ∗Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ The Conformal Window in SU(3) Yang-Mills Ethan T. Neil 1. Introduction With the Large Hadron Collider (LHC) about to come online, the bulk of particle theory re- search is focused on extending the Standard Model to the TeV-scale energies which will soon become experimentally accessible. A number of the theoretical models considered to describe LHC physics involve new strong dynamics, i.e. new interactions resembling the only known strong force, QCD. Models such as technicolor [1, 2], composite Higgs [3] and topcolor [4] are just a few examples of Standard Model extensions involving new strong interactions. If another strongly interacting theory does appear in nature, lattice gauge theory will provide PoS(LATTICE 2008)057 the ideal way to study it. Perturbation theory can provide only a limited understanding of a strongly coupled theory, and no other non-perturbative method is as mature and broadly applicable as lattice simulation. Even if no strongly coupled dynamics are revealed at the LHC, the study of more exotic Yang-Mills theories can be interesting in and of itself. Mapping out the parameter space of general Yang-Mills theories by varying the number of colors and light flavors would lead to a more solid theoretical understanding, which might open up new avenues of non-perturbative investigation into QCD or any strongly-coupled theory that might occur beyond the LHC. It is also possible that more exotic behavior may arise from strong dynamics. In particular, it is known that SU(N) gauge theory with enough light fermion flavors Nf develops an infrared conformal fixed point [5, 6]. Although not as directly applicable to LHC physics as the above examples, these theories are also of interest in the context of research into conformal field theory, e.g. [7, 8]. Although not truly conformal, these theories do show approximate scale invariance, and the ability to simulate such a theory on the lattice may offer new ways to study conformal behavior non-perturbatively. Furthermore, quantities derived from our running coupling measurement, such as the fixed-point coupling strength and the anomalous scaling dimension at the fixed point, may be of interest to model builders thinking about conformal behavior. Two of the most important qualities of QCD (Nf = 2 at relatively low energies) are asymp- totic freedom (the coupling strength vanishes at high energy/short distance) and confinement (the coupling strength diverges at low energy/long distance, i.e. there are no free states with color charge.) These properties are strongly dependent on Nf ; it is well known that for Nf > 16:5, SU(3) Yang-Mills is no longer asymptotically free [9]. These theories are too exotic for our current pur- poses. For Nf just below this critical value, the short-distance theory still resembles QCD, but in the infrared the coupling flows to a perturbative fixed point [6] - confinement is lost. c Clearly as we decrease the number of fermions, at some critical point 2 < Nf < 16:5 a transi- c tion must take place from a fixed-point theory to a confining theory. Since the theory for Nf > Nf c shows conformal behavior in the infrared, we refer to the range Nf < Nf < 16:5 as the confor- mal window. Although perturbation theory is useful near the top of the conformal window, as we decrease Nf the strength of the fixed-point coupling increases, eventually making the perturba- c tive expansion useless. Nf is thus unknown, and non-perturbative study is essential to accurately determine its value and the nature of the transition into the conformal window. 2. General approach and methods c As noted above, the two phases on either side of the transition at Nf are characterized by 2 The Conformal Window in SU(3) Yang-Mills Ethan T. Neil 4 !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ " ! #$ %& "! " "! ' PoS(LATTICE 2008)057 FIG. 2:Figure Cartoon 1: Cartoon showing showing the phases the insidephases and inside outside and outside of the cofonformal the conformal window, window, in terms in ofterms the of running the running coupling. The β-functioncoupling. (left) vanishesThe b-function in the ultraviolet (left) vanishes in both in cases, the ultravioletshowing asymptotic in both cases, freedom. showing The asymptotic difference is freedom. seen in the The infrared: a QCD-like theory confines, with the β-function appearing to diverge in the IR, while a theory in the conformal window develops an infrareddifference fixed pointis seen at ing = theg! infrared:. This is visiblea QCD-like in the theory running confines, coupling with itself the (right)b-function as a plateau. appearing to diverge in the IR, while a theory in the conformal window develops an infrared fixed point at g = g?. This is visible in the running coupling itself (right) as a plateau. !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ x4 expansion above are independent of renormalization T = L scheme, and given by xi confining and conformal2 2 behavior in the infrared; a simple way to distinguish them is thus toEk(η) b = 11 N (5) measure0 the running(4π)2 coupling− 3 f of the theory, and search for the presence or absence of an infrared ! " fixed point. This2 will be the38 focus of our lattice simulations. b = 102 N . (6) 1 (4π)4 − 3 f More concretely,! we begin with" the b-function of the theory, Asymptotic freedom is maintained so long as b is posi- dg0 2 tive, which holds for any Nf below 16.5. However, just 4 6 8 b(g) L = b0g + b1g + b2g + (2.1) below this value of Nf , the second coefficient≡ dLb1 is neg- ··· T =0 ative. Thus in the two-loop β-function, a zero devel- ops, whichwhich the encapsulates coupling constant the evolution will run of towards the coupling in the constant. The first two coefficients in the pertur- FIG. 3: Sketch of the Schr¨odinger Functional setup. Dirichlet infraredbative - an expansionIR fixed point above. areIn two-loop independent perturbation of renormalizationboundary scheme, conditions and in given the Euclidean by time direction are set theory this occurs for Nf near 8, but the value of the to impose a constant background chromoelectric field. coupling at the predicted fixed point is very strong; non- c perturbative study is the only way to truly determine2 Nf . 2 The direct non-perturbative measurementb0 of= the evolu-11 Nf 2 (2.2) (4p)2 −that3 the observable g matches onto the perturbative run- tion of the coupling constant is a simple way to distin- ning coupling. See figure III A for a sketch of the overall guish these two phases. A sketch of the β-function2 and setup.38 b1 = 102 Nf : (2.3) coupling constant evolution in the above cases is(4p shown)4 −The3 SF approach provides an alternative to the stan- in figure II B. dard Wilson loop method for measuring the running cou- WeAsymptotic choose to employ freedom the Schr¨odinger is maintained Functional so long as(SF)b is positive,pling strength, which one holds which for anyis freeN frombelow finite-size 16:5. effects definition of the running coupling in our measurement.0 f However, just below this value of N , the second coefficientsince the couplingb is negative. is measured Thus at in the the scale two-loop of the box, L. Dirichlet boundary conditions in Euclideanf time will be The presence1 of the box also lifts the zero modes of the used tob-function, impose a a constant zero develops, chromoelectric which the background coupling constantDirac willoperator, run towards making in it the possible infrared to simulate - an IR directly field; measuring the response of the system to variations fixed point. In two-loop perturbation theory this occursat forzeroN fermionf near 8, mass but the [11]. value This of theis crucial, coupling since non- in the background field will define the coupling strength. zero fermion masses would occlude the infrared behavior at the predicted fixed point is very strong; non-perturbative study is the only way to truly determine In particular, for the running coupling measurement the of the theory below that mass scale as the fermions are boundaryc gauge field values are set to be consistent with Nf . The direct non-perturbative measurement of thescreened evolution out. of the coupling constant is a simple the classical solution for a constant chromoelectric back- way to distinguish these two phases. A sketch of the b-functionA given lattice and coupling simulation constant must be evolution performed in at a fixed ground field, with strength parameterized by a dimen- 2 bare lattice coupling g0, which in turn fixes the scale of sionlessthe value aboveη.
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