We simulate SU(2) gauge theory with two massless Dirac fermions in the
adjoint representation. We calculate the running of the Schroedinger Functional
coupling and the renormalised quark mass over a wide range of length scales.
The running of the coupling is consistent with the existence of an infrared
fixed point (IRFP), and we find 0.07 < gamma < 0.56 at the IRFP, depending on
the value of the critical coupling.Comment: Talk presented at the XXVII International Symposium on Lattice Field
  Theory, July 26-31, 2009, Peking University, Beijing, China; 7 pages, 4
  figure

Bursa, Francis

Del Debbio, Luigi

Keegan, Liam

Pica, Claudio

Pickup, Thomas

English

arXiv

We simulate SU(2) gauge theory with two massless Dirac fermions in the adjoint representation. We calculate the running of the Schroedinger Functional coupling and the renormalised quark mass over a wide range of length scales. The running of the coupling is consistent with the existence of an infrared fixed point (IRFP), and we find 0.0

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     Edinburgh Research Explorer                                      Running of the coupling and quark mass in SU(2) with twoadjoint fermionsCitation for published version:Bursa, F, Del Debbio, L, Keegan, L, Pica, C & Pickup, T 2009, 'Running of the coupling and quark mass inSU(2) with two adjoint fermions', Paper presented at The XXVII International Symposium on Lattice FieldTheory, Beijing, China, 26/07/09 - 31/07/09. <http://arxiv.org/pdf/0910.2562.pdf>Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Early version, also known as pre-printPublisher Rights Statement:arXiv versionGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 04. Jan. 2021arXiv:0910.2562v1  [hep-ph]  14 Oct 2009 Running of the coupling and quark mass in SU(2)with two adjoint fermionsFrancis Bursa∗University of Cambridge, UKE-mail: fwb22@cam.ac.ukLuigi Del DebbioUniversity of Edinburgh, UKE-mail: luigi.del.debbio@ed.ac.ukLiam KeeganUniversity of Edinburgh, UKE-mail: liam.keegan@ed.ac.ukClaudio PicaUniversity of Edinburgh, UKE-mail: claudio.pica@ed.ac.ukThomas PickupUniversity of Oxford, UKE-mail: pickup@thphys.ox.ac.ukWe simulateSU(2) gauge theory with two massless Dirac fermions in the adjointrepresentation.We calculate the running of the Schrödinger Functional coupling and the renormalised quark massover a wide range of length scales. The running of the coupling is consistent with the existence ofan infrared fixed point (IRFP), and we find 0.07< γ < 0.56 at the IRFP, depending on the valueof the critical coupling.The XXVII International Symposium on Lattice Field TheoryJuly 26-31, 2009Peking University, Beijing, China∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/SU(2) with two adjoint fermions Francis Bursa1. IntroductionThe symmetry breaking in technicolor theories is communicated to the Standard Model by afurther interaction at some higher energy scaleM. There is a tension on the value ofM: on theone handM needs to be large so that FCNC interactions are suppressed, on the ther it needs to besmall to generate the heavier quark masses. The effective operat r for the quark masses is:L =1M2〈Φ〉ψ̄ψ , (1.1)whereψ indicates the quark field, andΦ the field which is responsible for electroweak symmetrybreaking. In traditional technicolor modelsΦ = Ψ̄Ψ is the chiral condensate of techniquarks. Thecoefficient in Eq. (1.1) is the chiral condensate at the scaleM:〈Ψ̄Ψ〉∣∣M = 〈Ψ̄Ψ〉∣∣Λ exp[∫ MΛdµµγ(µ)], (1.2)whereΛ is the technicolor scale. This suggests a possible way to resolve the tension onM: if γ isapproximately constant and large over a sufficiently long range in energies, then the condensate willbe enhanced. This scenario is known aswalking technicolor. Gauge theories with a large numberof fermions, or with fermions in higher–dimensional representations ofSU(N) [1], are candidates.These theories could have a genuine IR fixed point (IRFP), or simply be close to one.The existence of an IRFP is a difficult problem to address since it requires quantitative compu-tations in a strongly–interacting theory. Lattice simulations provide first–principle results that canhelp in determining the phenomenological viability of these models. A number of theories havebeen studied recently: SU(3) with 8, 10, 12 flavors of fermions in the fundamental representation,SU(3) with fermions in the sextet representation, and SU(2)with fermions in the adjoint represen-tation. Existing simulations of the Schrödinger functional h ve identified a possible fixed point inall the above–mentioned theories by noticing a flat behaviour of the running coupling.In this work we focus on SU(2) with 2 adjoint flavours, and compute the running coupling inthe SF scheme. We also compute the running of the mass, and extract the anomalous dimension.2. Basic formulationWe define the running couplingg2 non-perturbatively using the Schrödinger Functional method [2,3]. This is defined on a hypercubic lattice of sizeL, with boundary conditions chosen to impose abackground electric field on the system. The spatial link matrices att = 0 andt = L are set to:U(x,k)|t=0 = exp[ητ3a/iL] , U(x,k)|t=L = exp[(π −η)τ3a/iL] , (2.1)with η = π/4 [4]. The fermion fields obeyP+ψ = 0, ψP− = 0 att = 0 , P−ψ = 0, ψP+ = 0 att = L , (2.2)where the projectors are defined asP± = 1/2(1± γ0). The fermion fields also satisfy periodicspatial boundary conditions [5]. We use the Wilson plaquette gauge action, and Wilson fermionsin the adjoint representation, as implemented in Ref. [6].2SU(2) with two adjoint fermions Francis BursaThe coupling constant is defined asg2 = k〈∂S∂η〉−1(2.3)with k = −24L2/a2sin(a2/L2(π −2η)) chosen such thatg2 = g20 to leading order in perturbationtheory. This is a non–perturbative definition of the coupling which depends on only one scale,L.To measure the running of the quark mass, we calculate the pseudo calar density renormalisa-tion constantZP. Following Ref. [7],ZP is defined by:ZP(L) =√3 f1/ fP(L/2) , (2.4)where f1 and fP are the correlation functions involving the boundary fermion fieldsζ andζ :f1 = −1/12L6∫d3ud3vd3yd3z〈ζ′(u)γ5τaζ ′(v)ζ (y)γ5τaζ (z)〉 , (2.5)fP(x0) = −1/12∫d3yd3z〈ψ(x0)γ5τaψ(x0)ζ (y)γ5τaζ (z)〉 . (2.6)These correlators are calculated with the spatial link matrices att = 0 andL set to unity.We run directly atκc, determined through the PCAC massmPCAC(L/2), wheremPCAC(x0) =12(∂0 + ∂∗0 ) fA(x0)2 fP(x0)(2.7)andfA(x0) = −1/12∫d3yd3z〈ψ(x0)γ0γ5τaψ(x0)ζ (y)γ5τaζ (z)〉. (2.8)Here∂0 and∂ ∗0 are defined by∂0 f (x) = f (x+1)− f (x) and∂ ∗0 f (x) = f (x)− f (x−1). The corre-lators are calculated on lattices of sizeL with the spatial link matrices att = 0 andL set to unity.We defineκc by the point wheremPCAC vanishes on the 64 and 84 lattices, a linear extrapolationin a/L from these values, and the values for 164 lattices quoted in Ref. [8]. In practice we achieve|amPCAC| . 0.005. We check explicitly that there is no residual sensitivity to the small remainingquark mass by repeating some of our simulations atmPCAC∼ 0.02.3. Evidence for fixed pointsRecent lattice studies have focused on the running of the SF coupling, emphasizing the slowrunning of this quantity [9, 10, 11, 8]. These results have tobe interpreted with care. Lattice datacan find a range of energies over which no running is observed,but one cannot conclude that thisextends to arbitrarily large distances, as one would expectin the presence of an IRFP. On the otherhand, if the plateau has a finite extent,i.e. if the theory seems to walk, the behaviour of the runningcoupling depends on the scheme, and therefore the conclusions become less compelling.There are instances where the beta function of an asymptotically free theory is numericallysmall. This is the case of the theory considered in this work in the perturbative regime. In this case,even though the theory does not have a fixed point, the runningof the coupling is very slow. Highaccuracy is needed in order to resolve a “slow” running; therefore numerical studies of potential3SU(2) with two adjoint fermions Francis BursaIRFP need high statistics, and a robust control of systematics. In particular it is important toextrapolate to the continuum limit to eliminate lattice artefacts.Studies of the SF running coupling are a useful tool to exposethe possible existence of theoriesthat show a conformal behaviour at large distances. Howeverthe esults have to be interpretedwith care; they are unlikely to provide conclusive evidenceabout the existence of a fixed point bythemselves. A more convincing picture can emerge when they are combined with spectral studies.4. Results for the couplingWe have measured the couplingg2(β ,L) for a range ofβ ,L. Our results are plotted in Fig. 1.They are directly comparable to those of Ref [8], and agree within statistical errors. It is clear thatthe coupling is very similar for differentL/a at a given value ofβ , and hence that it runs slowly.2.0 3.0 4.0 5.0 6.0 7.0 8.0β0.01.02.03.04.05.0g2L/a=16L/a=12L/a=10 2/3L/a=9L/a=8L/a=6Figure 1: Data for the running coupling as computed from lattice simulations of the Schrödinger functional.In order to see how the coupling runs we define the step scalingfunctionσ(u,s) asσ(u,s) = g2(sL)|g2(L)=u (4.1)This is the continuum extrapolation ofΣ(u,s,a/L) which we calculate at variousa/L.We interpolate quadratically ina/L to find values ofg2(β ,L) at L = 9,1023 , which gives usdata for four steps of sizes = 4/3 for L → sL: L = 6,8,9,12; sL = 8,1023 ,12,16. Then for eachL we perform an interpolation inβ . We can then find estimates ofΣ(u,4/3,a/L) at anyu. Acontinuum extrapolation is then performed ina/L to give an estimate ofσ(u) ≡ σ(u,4/3). TheL = 6 data was found to have largeO(a) artifacts, and we have too fewL = 16 points to constrainthe interpolation functions, so neither are used in the continuum extrapolation.The resulting values forσ(u) can be seen in Fig. 2. The systematic errors from varying theinterpolation functions or the continuum extrapolation were significantly larger than the statisticalerrors. To quantify this, we recalculatedσ(u) with a range of different interpolation and extrapola-tion functions. The resulting extremal values ofσ(u) were used as upper and lower bounds on the4SU(2) with two adjoint fermions Francis Bursacentral value. The black error bars in Fig. 2 are determined ithis way, but using only a constantcontinuum extrapolation. These values are consistent witha fixed point in the regiong2 ∼ 2.0−3.2,as reported in Ref. [8]. The errors from also including the lin ar continuum extrapolation are muchlarger and mask any evidence for a fixed point, as also shown inFig. 2.0.0 1.0 2.0 3.0u0.981.001.021.041.061.081.10σ(u)/u1-loop2-loopFigure 2: The relative step–scaling functionσ(u)/u. The black circles use only a constant continuumextrapolation. The purple error bars include both constantand linear continuum extrapolations.5. Running massWe have measuredZP(β ,L) for a range ofβ ,L. We plot our results in Fig. 3, where we see aclear trend inZP as a function ofL at all β . The step scaling functionσP(u,s) is then defined as:σP(u,s) =ZP(sL)ZP(L)∣∣∣∣g2(L)=u(5.1)We extract this from a continuum extrapolation ofΣP(u,s,a/L).The method for calculatingσP(u) ≡ σP(u,4/3) is similar to that for calculatingσ(u). ZPconverges faster thang2 and we have better 164 data so here we use 4 points in our continuum ex-trapolations. Again the errors are dominated by systematics, in particular the choice of continuumextrapolation. We find good agreement with the 1-loop perturbative prediction.We cannot determine directly the running of the mass with scale since we observe no runningof the coupling within errors. However, we can define an estimator for the anomalous dimension,γ(u) = −ln |σP(u,s)|ln |s|, (5.2)which is equal to the anomalous dimension at an IRFP, and which we plot in Fig. 4. We see thatit is rather small over the range of interest; in particular,at g2 = 2.2, the benchmark value for theIRFP in [8], we haveγ = 0.114+78−35, and over the whole rangeg2 = 2.0− 3.2 consistent with anIRFP in [8], we find 0.07< γ < 0.56.5SU(2) with two adjoint fermions Francis Bursa2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0β0.100.200.300.400.500.600.700.800.901.00ZPL/a=6L/a=8L/a=9L/a=10 2/3L/a=12L/a=16Figure 3: Data for the renormalisation constantZP from lattice simulations of the Schrödinger functional.0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5u0.00.10.20.30.40.5γ(u)1-loopStatistical ErrorFigure 4: The mass anomalous dimensionγ(u). Red error bars use only a linear continuum extrapolation.Grey error bars include both constant and linear continuum extrapolations.6. ConclusionsIn these proceedings we have presented results for the running of the Schrödinger Functionalcouplingg2 and the mass anomalous dimensionγ .Our results for the running of the coupling are completely consistent with those of Ref. [8].Our statistical errors are larger; however, we have carriedout our analysis in a way that allows us totake the continnum limit with full control over the resulting systematic errors. Our results appearto show a slowing in the running of the coupling aboveg2 = 2 or so, and are consistent with the6SU(2) with two adjoint fermions Francis Bursapresence of a fixed point at somewhat higherg2. However, once we include the systematic errorsfrom the continuum extrapolation that our results no longergive any evidence for a fixed point.By contrast, we find that the behaviour of the anomalous dimension γ is much easier to es-tablish. We find a moderate anomalous dimension, close to the1-loop perturbative prediction,throughout the range ofβ explored. In particular, atg2 = 2.2, the benchmark value for the IRFPin [8], we find γ = 0.114+78−35. This value is much smaller than that required for phenomenology,which is typically of order 1-2. Such large values ofγ are clearly inconsistent with our results.The anomalous dimension is more vital than the running of2 for phenomenology; if it is notlarge then the presence or absence of walking behaviour becomes academic. Hence the implicationsof our measurement ofγ for minimal walking technicolor deserve to be studied carefully.The results presented here are preliminary, and the systematic errors need to be reduced tomake our conclusions more robust. Using larger lattices would make the continuum extrapolationsmore accurate, and it may also be necessary to use an improvedaction to reach the precisionrequired to show the existence of an IRFP or of walking behaviour. However this is very unlikelyto affect our phenomenologically most important result, namely thatγ is not large.References[1] D. Dietrich and F. Sannino,Conformal window of SU(N) gauge theories with fermions in higherdimensional representations, Phys. Rev. D75:085018, 2007.[2] M. Luscher, P. Weisz, and U. Wolff,A Numerical method to compute the running coupling inasymptotically free theories, Nucl. Phys. B359:221, 1991.[3] M. Luscher, R. Narayanan, P. Weisz, and U. Wolff,The Schrodinger functional: A Renormalizableprobe for nonAbelian gauge theories, Nucl. Phys. B384:168, 1992.[4] M. Luscher, R. Sommer, U. Wolff, and P. Weisz,Computation of the running coupling in the SU(2)Yang- Mills theory, Nucl. Phys. B389:247, 1993.[5] S. Sint and R. Sommer,The Running coupling from the QCD Schrodinger functional: A One loopanalysis, Nucl. Phys. B465:71, 1996.[6] L. Del Debbio, A. Patella, and C. Pica,Higher representations on the lattice: numerical simulations.SU(2) with adjoint fermions, arXiv:0805.2058.[7] S. Capitani, M. Luscher, R. Sommer, and H. Wittig,Non-perturbative quark mass renormalization inquenched lattice QCD, Nucl. Phys. B544:669, 1999.[8] A. Hietanen, K. Rummukainen, and K. Tuominen,Evolution of the coupling constant in SU(2) latticegauge theory with two adjoint fermions, arXiv:0904.0864.[9] T. Appelquist, G. Fleming, and E. Neil,Lattice Study of the Conformal Window in QCD-likeTheories, Phys. Rev. Lett. 100:171607, 2008.[10] Y. Shamir, B. 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Running of the coupling and quark mass in SU(2) with two adjoint fermions

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