- Email: [email protected]
Minimal technicolor on the lattice
Nuclear Physics A 820 (2009) 191c–194c www.elsevier.com/locate/nuclphysa
Minimal technicolor on the lattice Ari Hietanen a,b Jarno Rantaharju c Kari Rummukainen c,∗ Kimmo Tuominen a,d a Helsinki
Institute of Physics, University of Helsinki of Physics, Florida International University c Department of Physical Sciences, University of Oulu d Department of Physics, University of Jyv¨ askyl¨ a
b Department
Abstract We present results from a lattice study of SU(2) gauge theory with 2 flavors of Dirac fermions in adjoint representation. This is a candidate for a minimal (simplest) walking technicolor theory, and has been predicted to possess either an IR fixed point (where the physics becomes conformal) or a coupling which evolves very slowly, so-called walking coupling. In this initial part of the study we investigate the lattice phase diagram and the excitation spectrum of the theory.
Key words: technicolor, lattice gauge theory PACS: 11.15.Ha, 12.60.Nz
1. Introduction In technicolor theories the fundamental Higgs scalar is replaced with a composite scalar “meson”, consisting of two fermions, techniquarks, which interact through non-abelian technicolor gauge interaction. A very desirable feature in realistic technicolor theories, required in order to satisfy the experimental constraints from flavor changing neutral currents, is that the coupling constant evolves very slowly over a wide range of energy scales above the electroweak scale; so-called walking coupling, see [1] and references therein. For asymptotically free theories the β-function is negative for small couplings g 2 , and β(g 2 = 0) = 0. Walking behaviour is obtained if β(g 2 ) increases towards zero at some non-zero coupling, but not quite reaching it before decreasing again; see Fig. 1. On the ∗ Speaker
0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.01.047
192c
A. Hietanen et al. / Nuclear Physics A 820 (2009) 191c–194c
β
g
2
IR fixed point
g2 walking
QCD−like
ΛEW
E
Fig. 1. Left: An example of walking coupling. Right: the β-function which produces the walking coupling. Shown are also QCD-like β-function and β-function leading to a conformal IR fixed point.
other hand, if β(g 2 ) actually crosses zero, we obtain an IR fixed point, where the theory becomes conformal. Perturbative analysis indicates that SU(2) gauge field theory with 2 flavors of quarks in adjoint representation is a candidate for a theory with an IR fixed point [2]. Depending on non-perturbative dynamics (e.g. chiral symmetry breaking), the IR fixed point might turn into walking behaviour before the fixed point is reached. Because of its simplicity, this theory has the label “minimal walking technicolor”. Because the interesting physics — IR fixed point or walking behaviour — occurs at relatively large coupling, non-perturbative lattice simulations are required to resolve the true behaviour. In this work we report first results from simulations of SU(2) gauge with 2 flavors of lattice Wilson quarks in the adjoint representation. Here we investigate the lattice phase diagram and the mass spectrum of the theory; the evaluation of the evolution of the coupling constant is left for future work. This model has been recently studied by other groups [4,3]; the main difference is that here we use substantially larger volumes. 1 The first results have been summarized in more detail in [7], and full results will be published in [8]. 2. Phase diagram and mass spectrum 2 and the hopping parameter κ. κ The lattice action is parametrised with βL ≡ 4/gbare is related to the quark mass, but because of the lack of the chiral symmetry of the Wilson quark action, the true mass has to be defined e.g. through the axial Ward identity
mq = lim
t→∞
1 ∂t VPS , 2 VPP
(1)
where VPS and VPP are pseudoscalar-scalar and pseudoscalar-pseudoscalar current operators, respectively. In Fig. 2 we show the quark mass measurements as functions of κ for each of the the 5 values of βL used. The location where mq reaches zero defines the critical line κc (βL ). For small βL < ∼ 1.9 we observe an abrupt phase transition (on small lattices, not shown) into P and CP-breaking “Aoki phase” as mq becomes negative; this phase is an unphysical lattice artefact. At larger βL the behaviour remains regular as mq crosses zero, at least on finite volumes. The possible IR fixed point or walking behaviour naturally happens at the mq → 0 limit. 1 (See also [5,6] for recent lattice studies of SU(3) gauge with 2-index symmetric representation quarks, also a candidate theory for walking/conformal behaviour.
193c
A. Hietanen et al. / Nuclear Physics A 820 (2009) 191c–194c 1.2
Aoki phase β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
0.9
β=0 0.25
0.6
mq a
κ 0.2
mq = 0
0.3
mq > 0
0.15
β = infinity
0
0.12
0.15
0.18
κ
0.21
0.24
0.1 1
1.5
β
2
2.5
Fig. 2. Left: The quark mass as function of κ at different βL . The point where mq = 0 defines the critical line κc (βL ). Right: The phase diagram of the theory. The circles are measured points, and the thick line an approximate interpolation. The dashed borders of the unphysical Aoki phase are conjectural.
The physical excitation spectrum of the adjoint quark theory contains more states than the fundamental quark theory. In addition to 2-quark “mesons”, there are 3-quark “baryons” and bound states of a quark and a gluon, not present in the fundamental quark theory. Here we shall concentrate on the results from the pseudoscalar and vector meson measurements. The quark-gluon state is very noisy and we did not succeed to measure its mass. How should the existence of the IR fixed point be visible in the physical spectrum? As mq → 0, if the lattice coupling βL corresponds to the IR fixed point the asymptotic behaviour of all physical correlation functions show massless behaviour. This remains true for any coupling (in the domain of attraction of the IR fixed point), because the IR fixed point determines the IR physics. On the other hand, in case of walking coupling, at strong coupling we can have QCD-like chiral symmetry breaking. In Fig 3 the pseudoscalar (“π”) and vector (“ρ”) meson masses are presented. At small values of βL (strong bare coupling) the masses show clear pattern of chiral sym√ metry breaking: at small mq , pseudoscalar mass is proportional to mq , whereas the vector meson mass has finite intercept (as do the 3-quark spin 1/2 and 3/2 baryons). However, at larger βL (weak bare coupling) the masses behave approximately linearly in mq , compatible with conformal behaviour. In Fig. 4 the ratio of mπ /mρ is shown. This behaviour is compatible with the walking coupling scenario described earlier, with the almost-fixed-point near βL ≈ 2. It can also be compatible with true IR fixed point, provided that there is a phase transition between χSB at strong coupling, and conformal phase at weak coupling. However, one must interpret these results very cautiously: the behaviour is also compatible with QCD-like running coupling! We have compared the above results with those of SU(2) gauge and fundamental fermions, where there is χSB. However, at weak couplings the lattice volume simply becomes so small that the system appears deconfined, and the observed mass pattern is qualitatively very similar to the adjoint case, see Fig. 4. Thus, in order to resolve the issue a direct evaluation of the evolution of the coupling constant is required. Acknowledgement: JT and KR acknowledge the support of Academy of Finland grant 114371. The simulations were performed at the Center for Scientific Computing (CSC),
194c 3
3
2.5
2.5
2
2
mρ a
mπ a
A. Hietanen et al. / Nuclear Physics A 820 (2009) 191c–194c
1.5
1
β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
0.5
0
-0.3
0
0.3
mq a
0.6
0.9
β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
1.5
1
0.5
0
1.2
-0.3
0
0.3
mq a
0.6
0.9
1.2
1.1
1.1
1
1
0.9
0.9
0.8
0.8
mπ/mρ
mπ/mρ
Fig. 3. The pseudoscalar (“π”) mass (left) and the vector meson (“ρ”) mass (right) against the fermion √ mass mq . At small βL , mπ is proportional to mq , shown with a dotted line.
0.7 β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
0.6 0.5 0.4
0
0.3
0.6
mq a
0.9
0.7 β=1.7 β=2.5
0.6 0.5
1.2
0.4
0
0.3
0.6
mq a
0.9
1.2
Fig. 4. Pseudoscalar and vector meson mass ratios for the adjoint (left) and fundamental (right) fermions.
Finland, and at J¨ ulich supercomputer center (JSC). References [1] C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235 (2003) [Erratum-ibid. 390, 553 (2004)] [arXiv:hep-ph/0203079]. [2] F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005) [arXiv:hep-ph/0405209]; D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 72, 055001 (2005) [arXiv:hepph/0505059]. [3] L. Del Debbio, A. Patella and C. Pica, arXiv:0805.2058 [hep-lat]. [4] S. Catterall and F. Sannino, Phys. Rev. D 76 (2007) 034504 [arXiv:0705.1664 [hep-lat]]; S. Catterall, J. Giedt, F. Sannino and J. Schneible, arXiv:0807.0792 [hep-lat]. [5] Y. Shamir, B. Svetitsky and T. DeGrand, Phys. Rev. D 78 (2008) 031502 [arXiv:0803.1707 [hep-lat]]; arXiv:0809.2885 [hep-lat]; arXiv:0809.2953 [hep-lat]. [6] Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0809.4890 [hep-lat]; arXiv:0809.4888 [hep-lat]. [7] A. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, arXiv:0810.3722 [hep-lat]. [8] A. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, in preparation.
Minimal technicolor on the lattice Ari Hietanen a,b Jarno Rantaharju c Kari Rummukainen c,∗ Kimmo Tuominen a,d a Helsinki
Institute of Physics, University of Helsinki of Physics, Florida International University c Department of Physical Sciences, University of Oulu d Department of Physics, University of Jyv¨ askyl¨ a
b Department
Abstract We present results from a lattice study of SU(2) gauge theory with 2 flavors of Dirac fermions in adjoint representation. This is a candidate for a minimal (simplest) walking technicolor theory, and has been predicted to possess either an IR fixed point (where the physics becomes conformal) or a coupling which evolves very slowly, so-called walking coupling. In this initial part of the study we investigate the lattice phase diagram and the excitation spectrum of the theory.
Key words: technicolor, lattice gauge theory PACS: 11.15.Ha, 12.60.Nz
1. Introduction In technicolor theories the fundamental Higgs scalar is replaced with a composite scalar “meson”, consisting of two fermions, techniquarks, which interact through non-abelian technicolor gauge interaction. A very desirable feature in realistic technicolor theories, required in order to satisfy the experimental constraints from flavor changing neutral currents, is that the coupling constant evolves very slowly over a wide range of energy scales above the electroweak scale; so-called walking coupling, see [1] and references therein. For asymptotically free theories the β-function is negative for small couplings g 2 , and β(g 2 = 0) = 0. Walking behaviour is obtained if β(g 2 ) increases towards zero at some non-zero coupling, but not quite reaching it before decreasing again; see Fig. 1. On the ∗ Speaker
0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.01.047
192c
A. Hietanen et al. / Nuclear Physics A 820 (2009) 191c–194c
β
g
2
IR fixed point
g2 walking
QCD−like
ΛEW
E
Fig. 1. Left: An example of walking coupling. Right: the β-function which produces the walking coupling. Shown are also QCD-like β-function and β-function leading to a conformal IR fixed point.
other hand, if β(g 2 ) actually crosses zero, we obtain an IR fixed point, where the theory becomes conformal. Perturbative analysis indicates that SU(2) gauge field theory with 2 flavors of quarks in adjoint representation is a candidate for a theory with an IR fixed point [2]. Depending on non-perturbative dynamics (e.g. chiral symmetry breaking), the IR fixed point might turn into walking behaviour before the fixed point is reached. Because of its simplicity, this theory has the label “minimal walking technicolor”. Because the interesting physics — IR fixed point or walking behaviour — occurs at relatively large coupling, non-perturbative lattice simulations are required to resolve the true behaviour. In this work we report first results from simulations of SU(2) gauge with 2 flavors of lattice Wilson quarks in the adjoint representation. Here we investigate the lattice phase diagram and the mass spectrum of the theory; the evaluation of the evolution of the coupling constant is left for future work. This model has been recently studied by other groups [4,3]; the main difference is that here we use substantially larger volumes. 1 The first results have been summarized in more detail in [7], and full results will be published in [8]. 2. Phase diagram and mass spectrum 2 and the hopping parameter κ. κ The lattice action is parametrised with βL ≡ 4/gbare is related to the quark mass, but because of the lack of the chiral symmetry of the Wilson quark action, the true mass has to be defined e.g. through the axial Ward identity
mq = lim
t→∞
1 ∂t VPS , 2 VPP
(1)
where VPS and VPP are pseudoscalar-scalar and pseudoscalar-pseudoscalar current operators, respectively. In Fig. 2 we show the quark mass measurements as functions of κ for each of the the 5 values of βL used. The location where mq reaches zero defines the critical line κc (βL ). For small βL < ∼ 1.9 we observe an abrupt phase transition (on small lattices, not shown) into P and CP-breaking “Aoki phase” as mq becomes negative; this phase is an unphysical lattice artefact. At larger βL the behaviour remains regular as mq crosses zero, at least on finite volumes. The possible IR fixed point or walking behaviour naturally happens at the mq → 0 limit. 1 (See also [5,6] for recent lattice studies of SU(3) gauge with 2-index symmetric representation quarks, also a candidate theory for walking/conformal behaviour.
193c
A. Hietanen et al. / Nuclear Physics A 820 (2009) 191c–194c 1.2
Aoki phase β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
0.9
β=0 0.25
0.6
mq a
κ 0.2
mq = 0
0.3
mq > 0
0.15
β = infinity
0
0.12
0.15
0.18
κ
0.21
0.24
0.1 1
1.5
β
2
2.5
Fig. 2. Left: The quark mass as function of κ at different βL . The point where mq = 0 defines the critical line κc (βL ). Right: The phase diagram of the theory. The circles are measured points, and the thick line an approximate interpolation. The dashed borders of the unphysical Aoki phase are conjectural.
The physical excitation spectrum of the adjoint quark theory contains more states than the fundamental quark theory. In addition to 2-quark “mesons”, there are 3-quark “baryons” and bound states of a quark and a gluon, not present in the fundamental quark theory. Here we shall concentrate on the results from the pseudoscalar and vector meson measurements. The quark-gluon state is very noisy and we did not succeed to measure its mass. How should the existence of the IR fixed point be visible in the physical spectrum? As mq → 0, if the lattice coupling βL corresponds to the IR fixed point the asymptotic behaviour of all physical correlation functions show massless behaviour. This remains true for any coupling (in the domain of attraction of the IR fixed point), because the IR fixed point determines the IR physics. On the other hand, in case of walking coupling, at strong coupling we can have QCD-like chiral symmetry breaking. In Fig 3 the pseudoscalar (“π”) and vector (“ρ”) meson masses are presented. At small values of βL (strong bare coupling) the masses show clear pattern of chiral sym√ metry breaking: at small mq , pseudoscalar mass is proportional to mq , whereas the vector meson mass has finite intercept (as do the 3-quark spin 1/2 and 3/2 baryons). However, at larger βL (weak bare coupling) the masses behave approximately linearly in mq , compatible with conformal behaviour. In Fig. 4 the ratio of mπ /mρ is shown. This behaviour is compatible with the walking coupling scenario described earlier, with the almost-fixed-point near βL ≈ 2. It can also be compatible with true IR fixed point, provided that there is a phase transition between χSB at strong coupling, and conformal phase at weak coupling. However, one must interpret these results very cautiously: the behaviour is also compatible with QCD-like running coupling! We have compared the above results with those of SU(2) gauge and fundamental fermions, where there is χSB. However, at weak couplings the lattice volume simply becomes so small that the system appears deconfined, and the observed mass pattern is qualitatively very similar to the adjoint case, see Fig. 4. Thus, in order to resolve the issue a direct evaluation of the evolution of the coupling constant is required. Acknowledgement: JT and KR acknowledge the support of Academy of Finland grant 114371. The simulations were performed at the Center for Scientific Computing (CSC),
194c 3
3
2.5
2.5
2
2
mρ a
mπ a
A. Hietanen et al. / Nuclear Physics A 820 (2009) 191c–194c
1.5
1
β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
0.5
0
-0.3
0
0.3
mq a
0.6
0.9
β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
1.5
1
0.5
0
1.2
-0.3
0
0.3
mq a
0.6
0.9
1.2
1.1
1.1
1
1
0.9
0.9
0.8
0.8
mπ/mρ
mπ/mρ
Fig. 3. The pseudoscalar (“π”) mass (left) and the vector meson (“ρ”) mass (right) against the fermion √ mass mq . At small βL , mπ is proportional to mq , shown with a dotted line.
0.7 β=1.3 β=1.7 β=1.9 β=2.2 β=2.5
0.6 0.5 0.4
0
0.3
0.6
mq a
0.9
0.7 β=1.7 β=2.5
0.6 0.5
1.2
0.4
0
0.3
0.6
mq a
0.9
1.2
Fig. 4. Pseudoscalar and vector meson mass ratios for the adjoint (left) and fundamental (right) fermions.
Finland, and at J¨ ulich supercomputer center (JSC). References [1] C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235 (2003) [Erratum-ibid. 390, 553 (2004)] [arXiv:hep-ph/0203079]. [2] F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005) [arXiv:hep-ph/0405209]; D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 72, 055001 (2005) [arXiv:hepph/0505059]. [3] L. Del Debbio, A. Patella and C. Pica, arXiv:0805.2058 [hep-lat]. [4] S. Catterall and F. Sannino, Phys. Rev. D 76 (2007) 034504 [arXiv:0705.1664 [hep-lat]]; S. Catterall, J. Giedt, F. Sannino and J. Schneible, arXiv:0807.0792 [hep-lat]. [5] Y. Shamir, B. Svetitsky and T. DeGrand, Phys. Rev. D 78 (2008) 031502 [arXiv:0803.1707 [hep-lat]]; arXiv:0809.2885 [hep-lat]; arXiv:0809.2953 [hep-lat]. [6] Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0809.4890 [hep-lat]; arXiv:0809.4888 [hep-lat]. [7] A. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, arXiv:0810.3722 [hep-lat]. [8] A. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, in preparation.