PoS(LATTICE 2008)057 http://pos.sissa.it/ the theory changes fro confining to f N 8 and 12. = f N is varied. In particular, for sufficiently large ∗ f N [email protected] [email protected] [email protected] Speaker. conformal behavior in the infrared. We havestudy obtained of results the for SU(3) running Yang-Mills, coupling basedbetween on through these the the two Schrödinger Functional phases approach,staggered occurs fermion that between simulations the at 8 transition and 12 flavors. I will present the details of our Yang-Mills theory is known toflavors exhibit very different behavior as the number of light fermion ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

The XXVI International Symposium on LatticeJuly Field 14 Theory - 19, 2008 Williamsburg, Virginia, USA George T. Fleming Sloane Physics Laboratory, Yale University, New Haven,E-mail: CT 06511, United States Ethan T. Neil Thomas Appelquist Sloane Physics Laboratory, Yale University, New Haven,E-mail: CT 06511, United States The Conformal Window in SU(3) Yang-Mills Sloane Physics Laboratory, Yale University, New Haven,E-mail: CT 06511, United States PoS(LATTICE 2008)057 f c N (the > f asymp- confor- 5, SU(3) N . 5 a transi- . 16 Ethan T. Neil 16 > ] are just a few f < 4 5 as the confinement . f c N N 16 develops an infrared are characterized by < < f f c f N N N < f c ] and topcolor [ N 3 ] - confinement is lost. 6 2 at relatively low energies) are = ; it is well known that for f f N N 2 ]. These theories are too exotic for our current pur- 9 ], composite Higgs [ 2 , , i.e. new interactions resembling the only known strong 1 ]. Although not as directly applicable to LHC physics as the above is thus unknown, and non-perturbative study is essential to accurately 6 strong dynamics f c , gauge theory with enough light fermion flavors N 5 ) N ( just below this critical value, the short-distance theory still resembles QCD, but in (the coupling strength vanishes at high energy/short distance) and f the strength of the fixed-point coupling increases, eventually making the perturba- . Although perturbation theory is useful near the top of the conformal window, as we N f N ]. Although not truly conformal, these theories do show approximate scale invariance, and 8 , 7 As noted above, the two phases on either side of the transition at Clearly as we decrease the number of fermions, at some critical point 2 Two of the most important qualities of QCD ( With the Large Hadron Collider (LHC) about to come online, the bulk of particle theory re- If another strongly interacting theory does appear in nature, lattice gauge theory will provide It is also possible that more exotic behavior may arise from strong dynamics. In particular, determine its value and the nature of the transition into the conformal window. 2. General approach and methods mal window decrease tive expansion useless. shows conformal behavior in the infrared, we refer to the range Yang-Mills is no longer asymptotically free [ tion must take place from a fixed-point theory to a confining theory. Since the theory for totic freedom coupling strength diverges atcharge.) low These energy/long properties are distance, strongly dependent i.e. on there are no free states with color poses. For the infrared the coupling flows to a perturbative fixed point [ conformal fixed point [ examples, these theories are alsoe.g. of interest [ in the contextthe of ability research to into simulate conformal such field anon-perturbatively. theory, theory Furthermore, on quantities the derived lattice from may our offeras running new ways coupling the to measurement, fixed-point study such coupling conformal strength behavior be and of the interest anomalous to model scaling builders dimension thinking at about the conformal fixed behavior. point, may search is focused onbecome extending experimentally the accessible. StandardLHC Model A physics to involve number new the of TeV-scale the energies theoretical which models will considered soon to describe the ideal way to study it. Perturbationcoupled theory theory, and can no provide other only non-perturbative a method limitedsimulation. is understanding Even as of if mature a no and strongly strongly broadly coupled applicable dynamicsYang-Mills as theories are lattice revealed can at be the interesting LHC, in the andYang-Mills study theories of of more itself. by exotic Mapping varying out the the numbertheoretical parameter understanding, of space which colors of might general and open up light newQCD flavors avenues or would of any non-perturbative lead strongly-coupled investigation to into theory a that might more occur solid beyond the LHC. it is known that SU The Conformal Window in SU(3) Yang-Mills 1. Introduction force, QCD. Models such as technicolorexamples [ of Standard Model extensions involving new strong interactions. PoS(LATTICE 2008)057. 4 L let ects with ff ) η rared: a ( k L/a L E 5. IR . = =0 ! (2.2) (2.3) (2.1) T T " ' beyond 20 without below 16 Ethan T. Neil f L/a N . This is visible in e conformal window develops ? g erence is seen in the inf ff = lateau. g , which in turn fixes the scale of 2 0 is thus the only scale in matches onto the perturbative run- g 2 . Variation of the box size L g a i -function appearing to diverge in x ··· β TUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ + is negative. Thus in the two-loop -foldings. 8 e . 1 4 near 8, but the value of the coupling g b x !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRS 2 f  f b N  " N + f 6 3 N 38 g The SF approach provides an alternative to the stan- A given lattice simulation must be performed at a fixed fixed does not yield enough evolution of scale to get an -function and coupling constant evolution in 3 2 onformal window, in terms of the running coupling. The 1 − β b ning coupling. See figuresetup. III A for a sketch of thedard overall Wilson loop method forpling measuring strength, the running one cou- since which the is coupling is free measured from at the finite-size scale e of the box, the lattice spacing a accurate picture of thepractically running of we the cannot couplingthe constant; increase cost of thewe simulation becoming would prohibitive, like whereas over to many measure the evolution of the coupling bare lattice coupling The presence of theDirac box operator, also making liftsat it the zero zero possible fermion modeszero to of fermion mass simulate the masses [11]. directly wouldof occlude the the This infrared theory behavior is belowscreened out. that crucial, mass since scale non- as the fermions are boundary conditions into the impose Euclidean a time constant direction background are chromoelectric field. set that the observable FIG. 3: Sketch of the Functional setup. Schr¨odinger Dirich − + is positive, which holds for any ; the scale of the box 4 11 102 3 0 . 4 g c b f   0 L (6) (7) (5) showing asymptotic freedom. The di b 2 4 N ) ) " = 2 2 π π -function of the theory, is neg- is posi- 4 4 2 β ( ( 1 0 b b dL dg = = , the second coefficient L f 1 0 . -function and b b N 2 ! . ≡ β cient " ) 5. However, just " ffi . g f ( " N f β 3 N -function appearing to diverge in the IR, while a theory in th 38 β 2 3 . -function, a zero devel- − 2 β k . This is visible in the running coupling itself (right) as a p − below 16 g ! . In two-loop perturbation g f near 8, but the value of the 11 102 ≡ N is varied, = ! ! f η g η 2 4 d dS N ) ) , the second coe f -function (left) vanishes in the ultraviolet in both cases, showing asymptotic freedom. The 2 2 π π #$ %& β N (4 (4 ! = = . In two-loop perturbation theory this occurs for Cartoon showing the phases inside and outside of the conformal window, in terms of the running . The coupling is then measured to be 0 1 η b b IR fixed point We choose to employ the Schrödinger Functional (SF) definition of the running coupling in More concretely, we begin with the TUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRS is a constant of proportionality, which is set so . The direct non-perturbative measurement of the evolution of the coupling constant is a simple f c -function, a zero develops, which the coupling constant will run towards in the infrared - an k the above cases is shown in figure at the predicted fixed point is veryN strong; non-perturbative study is the onlyway way to to distinguish truly these determine two phases. A sketch of the fixed point our measurement. We work in a box of size Asymptotic freedom is maintained soHowever, long just as below this value of β the system, and our couplingtaken will in be the defined spatial at directions, this while same Dirichlet scale. boundary conditions Periodic in boundary Euclidean conditions time are will be used Figure 1: coupling. The difference is seen in the infrared: a QCD-like theory confines, with the confining and conformal behaviormeasure in the running the coupling infrared; offixed the a point. theory, simple and This will search way be for to the the focus distinguish presence of or them our absence lattice is of simulations. an thus infrared to The Conformal Window in SU(3) Yang-Mills the IR, while a theory inthe the running conformal coupling window itself develops (right) an as infrared a fixed plateau. point at which encapsulates the evolution of thebative coupling expansion above constant. are The independent first of two renormalization coefficients scheme, in and the given pertur- by We choose to employ the Functional Schr¨odinger (SF) -function (left) vanishes in the ultraviolet in both cases, coupling at the predictedperturbative fixed study point is is the only very way strong; to non- truly determine theory this occurs for inversely proportional tothe the strength response parameter of the action as guish these two phases. A sketch of the The direct non-perturbativetion measurement of of the the coupling evolu- constant is a simple way to distin- boundary gauge field valuesthe are classical set solution to for beground a consistent with constant field, chromoelectric with back- sionless strength value parameterized by a dimen- field; measuring the responsein of the the background system fieldIn to will particular, define variations the for coupling the strength. running coupling measurement the definition of theDirichlet running boundary coupling conditions in inused our Euclidean to measurement. time impose will be a constant chromoelectric background coupling constant evolution inin the figure II above B. cases is shown ative. Thus in the two-loop ops, which the couplinginfrared constant will - run an towards in the below this value of an infrared fixed point at tive, which holds for any expansion abovescheme, are and given independent by of renormalization Asymptotic freedom is maintained so long as β FIG. 2: Cartoon showing the phases inside and outside of the c where QCD-like theory confines, with the PoS(LATTICE 2008)057 (2.4) . The L Ethan T. Neil matches onto the ]. Trajectories are 2 . g 4 12 L let ects with ff ) η rared: a ( k L/a the scale of the box, L E = =0 at ! T T " ], version 6, with some customiza- ' 11 beyond 20 without . 2 L/a k g e conformal window develops 4 ≡ erence is seen in the inf , which are generally found to be negligible at small ff η 0 is performed to eliminate these errors. 2 dS d ], which is relatively cheap and simple to simulate with lateau. ) for a sketch of the overall setup. of the MD integrator is varied. This approach is subject = , which in turn fixes the scale of τ 2 13 2 τ 0 τ ∆ matches onto the perturbative run- g ∆ ( ∆ 2 . Variation of the box size g a is varied, i ]. This is crucial, since non-zero fermion masses would occlude x η TUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ 10 -foldings. e 4 x !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRS " The SF approach provides an alternative to the stan- A given lattice simulation must be performed at a fixed fixed does not yield enough evolution of scale to get an onformal window, in terms of the running coupling. The the lattice spacing a accurate picture of thepractically running of we the cannot couplingthe constant; increase cost of thewe simulation becoming would prohibitive, like whereas over to many measure the evolution of the coupling ning coupling. See figuresetup. III A for a sketch of thedard overall Wilson loop method forpling measuring strength, the running one cou- since which the is coupling is free measured from at the finite-size scale e of the box, The presence of theDirac box operator, also making liftsat it the zero zero possible fermion modeszero to of fermion mass simulate the masses [11]. directly wouldof occlude the the This infrared theory behavior is belowscreened out. that crucial, mass since scale non- as the fermionsbare are lattice coupling boundary conditions into the impose Euclidean a time constant direction background are chromoelectric field. set that the observable FIG. 3: Sketch of the Functional setup. Schr¨odinger Dirich . The coupling is then measured to be inversely proportional to the response of the η . c f Sketch of the Schrödinger Functional setup. Dirichlet boundary conditions in the Euclidean time (6) (7) (5) is a constant of proportionality, which is set so that the observable showing asymptotic freedom. The di N k " is neg- Simulations are carried out using the MILC code [ From a lattice simulation point of view, the SF approach provides an attractive alternative to We use the staggered fermion action [ is posi- ; on larger lattices, extrapolation to 1 a 0 b / b presence of the box alsodirectly lifts at the zero zero modes fermion of mass the [ Dirac operator, making it possible to simulate the infrared behavior of the theory below that mass scale as thetion. fermions Evolution are of screened the out. gauge(HMD) configurations technique, is accomplished with using the thetaken R hybrid to molecular algorithm be dynamics unit used length, to and the incorporate step size the fermions [ at zero fermion mass. However, due to the geometry of staggered fermions in combination with the standard Wilson loop methodfrom for finite-size measuring effects the running since coupling the strength, coupling one is which measured is free directly 3. Lattice simulation details to numerical integration errors of order L where perturbative running coupling. See figure Figure 2: to impose a constant chromoelectricvariations in background the field. background Measuring fieldcoupling will the measurement define response the the of boundary coupling the gauge strength.solution system field for In to values a particular, are constant for set chromoelectric thesionless to background running value be field, consistent with strength with parameterized theaction by classical as a the dimen- strength parameter direction are set to impose a constant background chromoelectric field. The Conformal Window in SU(3) Yang-Mills -function and ! . β cient

" 5. However, just " ffi .

f

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β 2

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.

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k . This is visible in the running coupling itself (right) as a p

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2 2 π π #$ %& N (4 (4 ! = = . The coupling is then measured to be 0 1 η b b IR fixed point TUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRS is a constant of proportionality, which is set so k We choose to employ the Functional Schr¨odinger (SF) -function (left) vanishes in the ultraviolet in both cases, coupling at the predictedperturbative fixed study point is is the only very way strong; to non- truly determine theory this occurs for the strength parameter inversely proportional to the response of the action as guish these two phases. A sketch of the The direct non-perturbativetion measurement of of the the coupling evolu- constant is a simple way to distin- In particular, for theboundary running gauge coupling field measurement valuesthe the are classical set solution to for beground a consistent with constant field, chromoelectric with back- sionless strength value parameterized by a dimen- coupling constant evolution inin the figure II above B. cases is shown definition of theDirichlet running boundary coupling conditions in inused our Euclidean to measurement. time impose willfield; be measuring a the constant responsein of the chromoelectric the background background system field to will define variations the coupling strength. ative. Thus in the two-loop infrared - an ops, which the coupling constant will run towards in the below this value of an infrared fixed point at tive, which holds for any expansion abovescheme, are and given independent by of renormalization Asymptotic freedom is maintained so long as FIG. 2:β Cartoon showing the phases inside and outside of the c where QCD-like theory confines, with the PoS(LATTICE 2008)057 ) 4) L ( 2 = (3.1) g a to link / L are adjusted Ethan T. Neil . We collect ) L , which in turn L . We can then 2 0 / , stepping from ( ) g 2 a L L . However, other g / / ) a step scaling a fixed does not yield L , are well fit by a pure -foldings. / ) a e ) a L , it is unsurprising that , L and integrator each step ( L u , so one would expect a 2 / , ) a g a 12 theory, however, shows 2 2 / with , , ( a 2 L u = ( a ( , Σ / f 2 O Σ and constant dropping ( L ) N lattice artifact is introduced when Σ ) L / ) L . Iterating this procedure allows us , and extract the correct continuum a is then doubled on each lattice, and a / ) a ( ( a L ]. L , 2 O O . This artifact is removed explicitly by 8 step is well-motivated by arguing that / u ( 13 , L a 2 2 g → ) + ( , an Σ a , each box size sL at four distinct values of ( β ± 2 fixed, Σ g L a 5 8 theory looks consistent with the confining phase, = ) = T = L f / a N , ) L . Variation of the box size . With . Keeping ( a a ) 2 indefinitely, capturing the evolution of the coupling constant / . The g L 2 from here forward. In practice, after a starting value , ( 3 20. boundary artifact operators which “stick out" from the Dirichlet T s 2 4 lattices are extremely large, to the point of non-analyticity; on ( ) = g to best fit the measured values for → a Σ beyond 20 without the cost of the simulation becoming prohibitive, = s ( → ··· L and averaging over the two [ a a O / ) / / a a L L L 2 and odd ± ( and 10 2 a L , g / L = 16 → 0 to recover the continuum value of T 3) lattice, ) → L → × ( 8 3 , 2 L 000 MD trajectories at each value of g , / 12 a → , in order to remove statistical bias and extract a good estimate of the running coupling. 6 , τ 8 To circumvent this problem, we use a systematic procedure known as A given lattice simulation must be performed at a fixed bare lattice coupling We will fix the step size Naïvely, staggered fermions possess discretization errors of Results are depicted in figure Since each gauge configuration yields only a single measurement of the running coupling, ∆ 000 to 80 , → whereas we would like to measure the evolution of the coupling over many enough evolution of scale to gettically an we accurate cannot increase picture of the running of the coupling constant; prac- together the results of manyrunning. simulations We with define the various lattice scales step scaling function to be over very large changes in scale. We determine 20 size fixes the scale of the lattice spacing extrapolate to step from 4 methods can be attempted. In particular, dropping the 4 showing no evidence of turnover to an infrared fixed point. The we attempt to measure thesimulating running at coupling both at the scale many updates are required to create statistically independent measurements of has been selected, there isto a yield tuning the step same in measurement the which measurement several of lattices the with different coupling on this larger lattice is exactly quadratic extrapolation in the lattice artifacts on the the smallest (4 boundaries will actually overlap. The remaining three values of are taken to define the edges of our systematic error band. there is no volume self-averaging, andsubstantial extracting a number measurement of with configurations. a reasonable Furthermore,the error order very requires of long a thousands autocorrelation of MD times trajectorieslocally are in while noted, the our worst on cases; observable as is the updating measured of on gauge fields the is scale done of the entire box The Conformal Window in SU(3) Yang-Mills the presence of Dirichlet boundary conditionson in lattices time with instead even of periodic, this forces us to simulate constant extrapolation, i.e. a weightedis average. found The to difference in beextrapolation choice the methods of (quadratic dominant extrapolation to method source all four of values systematic of error in our simulations; therefore these two PoS(LATTICE 2008)057 10 f c ], N 19 8 – ]. The 15 14 Ethan T. Neil 10, which is 6 # 0 lies outside the = L " f ]. f L ! N 16 N Log 4 Yang-Mills, as in [ ]. ) 3 21 2 ( 2-loop univ. 3-loop SF ]. SU 20 is possible with Wilson fermions; a / 12 is quite weak. If this continues to L 0 = 8 6 4 2 f 16 14 12 10 % N L $ 8 (right) in 2 8 indicates that this value of to the continuum extrapolation, constraining g ) = = L f 35 f / 6 N N a , u operators. Additionally, we can switch to the Wilson- dependence by using the clover-improved action, and , 30 ) 2 L a ( / ( Σ a O 25 12 (left) and 10 would require taking a fractional power of the fermion deter- = # = f 0 20 L f N " L N ! Log 15 10 2-loop univ. 3-loop SF 5 Running coupling at 12, a reduced error band should admit a measurement of the anomalous dimension of the 0 = 2 8 6 4 The logical next step is a similar measurement of the running coupling at Work is still ongoing to refine these simulations and their analysis. The systematic error bands In conclusion, we have shown evidence that the lower boundary of the conformal window Furthermore, the results as depicted are consistent with 3-loop perturbation theory, and indeed Wilson fermions are inherently much more expensive to simulate with than staggered, but this f 16 14 12 10 % L N $ 2 ongoing. To continue simulating directly atuse of zero staggered quark fermions mass, at we mustminant, switch which can to be Wilson problematic fermions; if the fermion mass is set to zero [ lies between 8 and 12 flavors.and This in is contrast in with agreement with the most previous perturbative lattice estimates, results e.g. of [ Iwasaki et al. [ shown are quite large, andat reducing them would increase confidence in our results. In particular the coupling strength of the IR fixed point observed at will be offset in several ways. First of all, simulation at odd conformal window. clear evidence of an IRFP. Indeed, thisconformal theory behavior represents outside the of first supersymmetry. non-perturbative measurement of coupling constant as it approachesworking the with conformal fixed and point, near-conformal a theories. quantity which may interest model-builders 4. Conclusion and outlook this will allow us to add more values of Figure 3: The Conformal Window in SU(3) Yang-Mills systematic error band (grey)in is the determined text. by variation The of lack continuum of extrapolation an method, observed as fixed described point at be the case as theperturbation error theory band is is valid all narrowed, the it way would to lend the strength bottom of to the the conformal claim window by [ Gardi et al. that it better. We will also attempt to reduce perturbative counterterms for boundary oriented Chroma code base, whichhybrid is Monte very Carlo, optimized which and should offers further improved diminish algorithms simulation like cost. rational g PoS(LATTICE 2008)057 , , , Phys. 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Renken, and R. L. Sugar. Hybrid molecular dynamics [15] Thomas Appelquist, Andrew G. Cohen, and Martin Schmaltz. A new constraint on strongly coupled [18] V. A. Miransky and Koichi Yamawaki. Conformal phase transition in gauge theories. [20] Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie. Phase[21] structure of Stephen lattice R. QCD Sharpe. for Rooted general staggered fermions: Good, bad or ugly? [16] Einan Gardi and Georges Grunberg. The conformal window in QCD and[17] supersymmetric QCD. Thomas Appelquist, John Terning, and L. C. R. Wijewardhana. Postmodern technicolor. [10] Rainer Sommer. Non-perturbative qcd: renormalization, The Conformal Window in SU(3) Yang-Mills References