Low energy spectra in many flavor QCD with Nf=12 and 16
Yasumichi Aoki, Tatsumi Aoyama, Masafumi Kurachi, Toshihide Maskawa, Kei-ichi Nagai, Hiroshi Ohki, Akihiro Shibata, Koichi Yamawaki, Takeshi Yamazaki
aa r X i v : . [ h e p - l a t ] N ov Low energy spectra in many flavor QCD with N f = Yasumichi Aoki a , Tatsumi Aoyama a , Masafumi Kurachi a , Toshihide Maskawa a ,Kei-ichi Nagai a , Hiroshi Ohki ∗ a , Akihiro Shibata b , Koichi Yamawaki a andTakeshi Yamazaki a LatKMI Collaboration a Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), NagoyaUniversity, Nagoya 464-8602, Japan b Computing Research Center, High Energy Accelerator Research Organization (KEK), Tsukuba305-0801, JapanE-mail: [email protected]
We present our result of the many-flavor QCD. Information of the phase structure of many-flavorSU(3) gauge theory is of great interest, since the gauge theories with the walking behavior nearthe infrared fixed point are candidates of new physics for the origin of the dynamical electroweaksymmetry breaking. We study the SU(3) gauge theories with 12 and 16 fundamental fermions.Utilizing the HISQ type action which is useful to study the continuum physics, we analyze thelattice data of the mass and the decay constant of the pseudoscalar meson and the mass of thevector meson as well at several values of lattice spacing and fermion mass. The finite size scalingtest in the conformal hypothesis is also performed. Our data is consistent with the conformalscenario for N f =
12. We obtain the mass anomalous dimension g m ∼ . − .
5. An update of N f =
16 study is also shown.
The 30 International Symposium on Lattice Field Theory - Lattice 2012,June 24-29, 2012Cairns, Australia ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ow energy spectra in many flavor QCD with N f =
12 and 16
Hiroshi Ohki
1. Introduction
There has been a renewed interest in the study of QCD with large number of the masslessfermions in the fundamental representation (“large N f QCD”) in the context of walking technicolorhaving approximate scale invariance in the infrared (IR) region and large anomalous dimension g m ≃ N f QCD: The perturbative two-loop beta function predicts a non-trivial infrared fixed point a ∗ (0 < a ∗ < ¥ ) in the range of 9 ≤ N f ≤
16 in the asymptotically free SU(3) gauge theory [2, 3]. As apowerful tool of a nonperturbative study, one uses lattice QCD simulations, which can in principledetermine the phase structure of the SU(3) gauge theories with various number of fermions. Inaddition to the pioneering works [4, 5, 6], there are many lattice works in the large N f QCD in therecent years. In particular, the system of the N f =
12 has been widely investigated by the latticeapproach, such as running coupling, lattice phase diagram, low-energy spectra and so on. (See, fora review, Ref. [7].)We investigate the 12 and 16-flavor SU(3) gauge theories using a variant of the highly im-proved staggered quark (HISQ) action [8] to reduce the discretization error. We study severalbound-state masses such as the pseudoscalar meson p and vector meson r as well as the decayconstant of p , by varying the fermion bare mass m f . In this work, we introduce a quantity for thescaling test of the conformal hypothesis with finite volume in the analyses for N f =
12. Using thisquantity, we can analyze the data without any assumption of the fitting form in the scaling test.We also discuss possible finite size and mass corrections in the scaling. We find that our results of N f =
12 are consistent with hyperscaling with g = . − .
5. These results of N f =
12 have alreadybeen published in Ref. [9].Besides the N f =
12 theory, we also simulate the N f =
16 theory with the same setup. Sincethis theory is expected to deeply reside in the conformal phase, it is helpful to understand conformalsignals from numerical simulations. The preliminary results of this theory are also shown. N f = We use a version of the HISQ [8] action for many-flavor simulations but without the tadpoleimprovement and the mass correction term for heavy fermions. Gauge configurations are generatedby HMC algorithm using MILC code ver.7 with various parameter sets for the fermion mass m f ,volume and the bare coupling b = / g . We calculate several bound-state masses, such as thepseudoscalar meson p and vector meson r , and the decay constant of p . If the theory is in theconformal window, the hadron mass M H and p decay constant F p obey the conformal hyperscaling M H (cid:181) m g ∗ + f , F p (cid:181) m g ∗ + f , (2.1)where g ∗ denotes the mass anomalous dimension at the IR fixed point. On the other hand, if thetheory is in the phase of chiral symmetry breaking, the leading fermion mass dependence of p is M p (cid:181) m f , F p = c + c m f , (2.2)with c =
0, and the vector meson mass does not vanish in the chiral limit.The spectra obtained in our lattice simulation will be tested against these two hypotheses inthis work. 2 ow energy spectra in many flavor QCD with N f =
12 and 16
Hiroshi Ohki
For a primary analysis dimensionless ratios, F p / M p and M r / M p , are plotted against aM p inFig. 1. The left panel plots F p / M p on the largest two volumes at the two bare gauge couplings b = . b = . F p / M p tends to be flat forsmaller p masses, which shows clear contrast to ordinary QCD case. The behavior is consistentwith the hyperscaling in Eq. (2.1). The M p dependence of the ratio at the larger mass can berealized by the correction to the hyperscaling which may be different from one quantity to another.For b = M r / M p shown in the right panel of Fig. 1. The flattening is observedfor b = . F p / M p . In this case b = F p at b = . b = aM p can be madepossible if the M p in the physical unit is larger (thus the correction is no longer negligible) for b = i.e. , the lattice spacing decreases as b increases. In that case the physical volume is smallerfor b =
4, which gives a reason for the volume effect observed only for b =
4. Actually a crudeanalysis of two lattice spacing matching shows the result a ( b = . ) > a ( b = ) which is consistentwith being in the asymptotically free domain. p F p / M p L=24L=30 p M r / M p L=24L=30
Figure 1:
Dimension-less ratios F p / M p and M r / M p as functions of aM p for N f =
12 at b = . . In the conformal window with finite masses and volume, the renormalization group analysistells us that the scaling behavior for low-energy spectra which should obey the universal scalingrelations as x p ≡ LM p = f p ( x ) , x F ≡ LF p = f F ( x ) , (2.3)where the subscript p distinguishes the bound state, p = p or r in this study. The product ofbound state mass or decay constant and linear system size falls into a function of a single scalingvariable x = L · m + g ∗ f , where g ∗ is the mass anomalous dimension at the IR fixed point. We shall callthe scaling relation the finite-size hyperscaling (FSHS). While the forms of the scaling functions For reviews, see e.g. [10, 11]. ow energy spectra in many flavor QCD with N f =
12 and 16
Hiroshi Ohki xp g =0.1 0 2 4 6 8 10 12x051015202530 xp g =0.4 0 2 4 6 8 10 12x051015202530 xp g =0.7 Figure 2: x p is plotted as a function of the scaling variable x for g = .
1, 0 .
4, and 0 . N f =
12 at b = .
7. An alignment is seen for g ∼ . f p ( x ) are unknown in general, the asymptotic form should be f ( x ) ∼ x at large x because it mustreproduce the hyperscaling relation Eq. (2.1) in large volumes.Now we examine whether our data for the bound state masses and decay constant obey FSHS.To visualize how the scaling works we show x p as functions of x = L · m / ( + g ) f for several values of g in Fig. 2. It is observed that the data align well with around g = .
4, while they become scatteredfor g away from that value. This indicates the existence of possible FSHS with g ∼ .
4. A similaralignment is observed for x F as well. In this case one finds the optimal scaling at around g ∼ . P ( g ) for an observable p asfollows. Suppose x j be a data point of the measured observable p at x j = L j · m / ( + g ) j and dx j be the error of x j . j labels distinction of parameters L and m f . Let K be a subset of data points { ( x k , x k ) } from which we construct a function f ( K ) ( x ) that represents the subset of data. Then theevaluation function is defined as P ( g ) = N (cid:229) L (cid:229) j K L (cid:12)(cid:12) x j − f ( K L ) ( x j ) (cid:12)(cid:12) | dx j | , (2.4)where L runs through the lattice sizes we have, the sum over j is taken for a set of data points thatdo not belong to K L which includes all the data obtained on the lattice with size L . N denotesthe total number of summation. Here we choose for the function f ( K L ) a linear interpolation ofthe data points of the fixed lattice size L for simplicity, which should be a good approximation of x for large x . In the evaluation function Eq. (2.4), the data points need to be taken for a rangeof x = L · m / ( + g ) f in which there is an overlap of available data for all volumes, L =
18, 24, and30 within the range [ x min , x max ] . We take the value of x min ( x max ) as the smallest (largest) m f forthe largest (smallest) volume L in our simulation parameters. Note, however, we may need toincorporate some neighboring data outside this range to obtain the interpolated value f ( K ) ( x ) bythe spline functions.The evaluation function for all the quantities, M p , M r and F p , is plotted in the left panel ofFig. 3. A clear minimum is observed at which the optimal alignment of the data is achieved. It isnoted that the value of P ( g ) is O ( ) at the minimum. The systematic error due to the ambiguity ofthe interpolation is estimated by the difference of the optimal g ’s obtained with linear and quadraticspline interpolations. The comparison of these P ( g ) ’s is also shown in the left panel of Fig. 3. Theminima for the quadratic spline interpolation appear approximately at the same place as those forthe linear one. It is found that this systematic error is always smaller than the statistical error. Theother uncertainties due to the finite size and mass effects are estimated by the variations of optimal4 ow energy spectra in many flavor QCD with N f =
12 and 16
Hiroshi Ohki g P M p (linear)M p (quadratic)F p (linear)F p (quadratic)M r (linear)M r (quadratic) g M p ( b =4.0)M p ( b =3.7)M r ( b =3.7)M r ( b =4.0)F p ( b =3.7)F p ( b =4.0) Figure 3: (Left) The g dependence of the evaluation function P for M p , F p , and M r at b = . P at each of g where the three volumes and full range of x for the dataare considered. The solid and dashed curves show the results of P ( g ) with the interpolation functions f ( x ) bythe linear and quadratic functions, respectively. (Right) The results of the values of g for three observablesat two b are summarized, where the statistical and systematic errors are added in quadrature. g with respect to the change of both the x-range and L used in the analyses. The results with all theerrors added in quadrature are summarised in the right panel of Fig. 3. The details of the analysisare shown in Ref. [9]. All the results are consistent with each other within 2 s level, except for g from F p at b = N f =
12 theory are reasonably consistent with the FSHS. The resulting mass anomalousdimensions is 0 . ≤ g ∗ ≤ . c = N f (cid:16) M p p F p / √ (cid:17) is very large in the region we simulated, which is evaluated as c ∼
39 atthe smallest M p using the value of F p in the chiral limit. With this large c , we could not consis-tently analyze the data based on the ChPT. Further efforts would be required to arrive at a decisiveconclusion. N f = In the previous report [12] we presented that the result of the x p is consistent with the FSHS,but the optimal g decreases as b increases in b ≤ .
5. Furthermore the results of the g is muchlarger than the perturbative result, g ∼ . b dependence of g in larger b regionthan the ones we simulated in the previous report.Using the same simulation setup as in N f =
12, we perform simulations at several values of b adding to the previous work, such as 5 and 12, on various spatial volumes, L = , , ,
24 and30. The range of the fermion mass is 0 . ≤ m f ≤ .
2, and the typical length of the trajectory isroughly 1000.The plots in Fig. 4 show that for b = x p aligns well as a function of x usingan optimal value of the g as in the N f =
12 case. The value, however, largely depends on b .At the highest b the g is still five times larger than the perturbative result, although the result atthis b would include large systematic error coming from finite volume. Fig. 5 shows the scatter5 ow energy spectra in many flavor QCD with N f =
12 and 16
Hiroshi Ohki x p b =3.15 g ~0.415 x p b =5 g ~0.24 x p b =12 g ~0.13 Figure 4: x p in the 16 flavors at b = . -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2Re-0.2-0.15-0.1-0.0500.050.10.150.2 I m XYZ b= 5.0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2Re-0.2-0.15-0.1-0.0500.050.10.150.2 I m XYZ b= 8.0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2Re-0.2-0.15-0.1-0.0500.050.10.150.2 I m XYZ b=12.0
Figure 5:
Scatter plot of the Polyakov loops in the spacial directions. plots of Polyakov loop in the spatial directions with fixed bare mass m f = . L = , T =
32. As shown here, at the highest b , the Polyakov loop has a non-zero value associatedwith the center symmetry breaking. On the other hand, at the lower b region, this value decreaseswith b . We consider that this symmetry breaking at the higher b occurs due to the small physicalvolume, so that to reduce finite volume effects we would need much larger lattice size than L = b =
12. We will continue investigations with larger volumes at higher b region to obtain thecorrect value of the g at the vicinity of the infrared fixed point.
4. Summary and outlook
We have studied the SU(3) gauge theories with the fundamental 12 and 16 fermions using aHISQ type staggered fermion action, For the 12-flavor case, we attempt to determine the phaseof this theory through the analysis of M p , M r and F p . Our present data is consistent with theconformal hypothesis. The mass anomalous dimension, g ∗ at the infrared fixed point was estimatedthrough the (finite-size) hyperscaling analysis. Our result is g ∗ ∼ . − .
5, which is not as bigas g ∗ ∼ c is much larger than one even at the smallest M p . Wecould not consistently analyze the data based on the chiral expansion. A possibility of the chiralbroken phase in N f =
12 is not excluded yet. More detailed analyses with more data at largervolume and lighter mass would be required. For the 16-flavor case, the pion mass data exhibit thescaling which is consistent with the conformal scenario, while the obtained value of the g is much6 ow energy spectra in many flavor QCD with N f =
12 and 16
Hiroshi Ohki bigger than the perturbative result. To obtain the g at the infrared fixed point, further study of the g , especially volume dependence would be required. Another series of the study in our project hasbeen reported for the test of the walking behavior in N f = Acknowledgments
Numerical calculations have been carried out on the cluster system j at KMI, Nagoya Univer-sity. This work is supported in part by JSPS Grants-in-Aid for Scientific Research (S) No. 22224003,(C) No. 21540289 (Y.A.), (C) No. 23540300 (K.Y.), (C) No. 24540252 (A.S.) and Grants-in-Aidof the Japanese Ministry for Scientific Research on Innovative Areas No. 23105708 (T.Y.). References [1] K. Yamawaki, M. Bando and K. -i. Matumoto, Phys. Rev. Lett. , 1335 (1986).[2] W. E. Caswell, Phys. Rev. Lett. (1974) 244.[3] T. Banks and A. Zaks, Nucl. Phys. B (1982) 189.[4] Y. Iwasaki, K. Kanaya, S. Sakai and T. Yoshie, Phys. Rev. Lett. (1992) 21.[5] F. R. Brown, H. Chen, N. H. Christ, Z. Dong, R. D. Mawhinney, W. Schaffer and A. Vaccarino, Phys.Rev. D (1992) 5655 [hep-lat/9206001].[6] P. H. Damgaard, U. M. Heller, A. Krasnitz and P. Olesen, Phys. Lett. B (1997) 169[hep-lat/9701008].[7] J. Giedt, plenary talk at Lattice 2012, PoS LATTICE (2012) 006; E. T. Neil, PoS LATTICE , 009 (2011) [arXiv:1205.4706 [hep-lat]], and references therein.[8] E. Follana et al. [HPQCD and UKQCD Collaborations], Phys. Rev. D (2007) 054502[hep-lat/0610092].[9] Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. -i. Nagai, H. Ohki, A. Shibata, K. Yamawaki,T. Yamazaki, (LatKMI collaboration), Phys. Rev. D (2012) 054506 [arXiv:1207.3060 [hep-lat]].[10] T. DeGrand and A. Hasenfratz, Phys. Rev. D (2009) 034506 [arXiv:0906.1976 [hep-lat]].[11] L. Del Debbio and R. Zwicky, Phys. Rev. D (2010) 014502 [arXiv:1005.2371 [hep-ph]].[12] Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. -i. Nagai, H. Ohki, A. Shibata, K. Yamawaki,T. Yamazaki, (LatKMI collaboration), PoS LATTICE , 080 (2011) [arXiv:1202.4712 [hep-lat]].[13] Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. -i. Nagai, H. Ohki, A. Shibata, K. Yamawaki,T. Yamazaki, (LatKMI collaboration), talk at Lattice 2012, PoS LATTICE (2012) 035.[14] Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. -i. Nagai, H. Ohki, A. Shibata, K. Yamawaki,T. Yamazaki, (LatKMI collaboration), Phys. Rev. D (2012) 074502 [arXiv:1201.4157 [hep-lat]];talk at Lattice 2012, PoS LATTICE (2012) 059.(2012) 059.