Fermions in higher representations. Some results about SU(2) with adjoint fermions.

Transcription

1 Fermions in higher representations. Some results about SU(2) with adjoint fermions. Agostino Patella Swansea University with L. Del Debbio, C. Pica arxiv: [hep-lat] Lattice08, Williamsburg 18/7/2008

2 Outline 1 Motivations Why higher representations? 2 The HiRep code The HiRep code 3 SU(2) with n f = 2 adjoint Conformal point? The parameters The chiral limit Troubles at small m at fixed lattice spacing Extracting the masses from correlators Results 4 Conclusions

3 Outline 1 Motivations Why higher representations? 2 The HiRep code The HiRep code 3 SU(2) with n f = 2 adjoint Conformal point? The parameters The chiral limit Troubles at small m at fixed lattice spacing Extracting the masses from correlators Results 4 Conclusions

4 Motivations: Why higher representations? Technicolor models. SU(2) gauge theories + fermions in the symmetric two-index representation Orientifold planar equivalence. SU(3) + fund. fermions SU(N) + 2AS fermions SU( ) + Adj fermions Softly-broken SYM. SU(N) gauge theories + one Majorana fermion in the adjoint representation

5 Outline 1 Motivations Why higher representations? 2 The HiRep code The HiRep code 3 SU(2) with n f = 2 adjoint Conformal point? The parameters The chiral limit Troubles at small m at fixed lattice spacing Extracting the masses from correlators Results 4 Conclusions

6 The HiRep code Wilson action + Wilson fermions. Standard HMC/RHMC algorithm. Second order Omelyan integrator for the molecular dynamics evolution, with different time steps for the gauge and fermion actions. Link update implemeted by left multiplication of a unitary matrix that is a second-order approximation for exp (iπ t). Even/odd preconditiong for the Dirac operator. Fermions in the representation R (fund, 2AS, 2S, Adj). Dψ(x) = ψ(x) 1 κ X µ n o (1 γ µ)u R (x, µ)ψ(x + µ) + (1 + γ µ)u R (x µ, µ) ψ(x µ) d dτ U(x, µ) = iπa (x, µ)tf a U(x, µ) H = 1 X π a (x, µ) 2 β X P µν(x) + X φ (x)[d R 4 N D R s] 1 φ(x) x,µ x,µ<ν x dh f dτ = i X π a (x, µ)tr R {TR a F f [U R ](x, µ)} a,x,µ 1 d 2 dτ πa (x, µ) + tr F [itf a Fg(x, µ)] + tr R[iTR a F f (x, µ)] = 0

7 Outline 1 Motivations Why higher representations? 2 The HiRep code The HiRep code 3 SU(2) with n f = 2 adjoint Conformal point? The parameters The chiral limit Troubles at small m at fixed lattice spacing Extracting the masses from correlators Results 4 Conclusions

8 Conformal point? m Conf ChSB ChS beta Catteral, Giedt, Sannino, Schneible, hep-lat/ next talk by Hietanen

9 The parameters β = 2.0 lattice V κ am 0 N traj P τ T2-A (14) 2.9(0.4) T2-A (16) 3.1(0.5) T2-A (18) 3.1(0.5) T2-A (20) 6.0(1.2) T2-A (37) 12.0(3.6) T2-A (50) 22.3(9.3) T2-A (55) 48.3(23.3) T2-A (58) 40.7(16.3) T2-A (45) 2.7(1.2) T2-B (42) 5.8(3.6) T2-B (56) 9.0(9.6) T2-B (53) 4.2(2.6) T2-B (73) 2.6(1.8) T2-B (58) 13.6(11.5)

10 The chiral limit Wilson fermions explictly break the chiral symmetry for each value of κ. The chiral point is fine-tuned by requiring that the Ward identities for the chiral symmetry are recovered. ψγ 5 ψ(x) µ ψγµγ 5 ψ(y) = 2m ψγ 5 ψ(x) ψγ 5 ψ(y) 1 am A κ 1 «κ c In the chiral point m = 0 and in the continuum limit, we assume that the chiral symmetry is spontaneously broken. Then the lightest PS meson is massless and for small values of m PS, the χpt is valid. M PS 4πF PS 1 am PS B am F PS F PS (0) + Cm M V M V (0) + Dm We want to check this assumption. We need to go to small PCAC masses but we need to be careful in this region.

11 Troubles at small m at fixed lattice spacing SU(4) broken by m, a sp. broken in Aoki phase SO(4) U(1) U(1) In the Aoki phase, flavour is spontaneously broken. Four Goldstone bosons are expected in this case. The transition to the Aoki phase is expected around the chiral limit in a width a m (aλ) 3. In the Aoki phase, exact zero modes of the Dirac operator (and instability of the algorithm) are expected. At finite volume, the phase transition becomes a wide cross-over. Thus, we can have a region of stability of the algorithm in which the measured observables are highly sensitive to lattice artifacts. Safe chiral limit We need to check the stability of our results close to the chiral point, by increasing the volume and reducing the lattice spacing (the width of the distribution of the lowest Dirac eigenvalue shrinks as a/ V and the Aoki phase width shrinks as a 3 ).

12 Extracting the masses from correlators γ 0 γ 5 γ am PS t/a h(t + 1, M) h(t, M) = Cγ 5,γ 5 (t + 1) C γ5,γ 5 (t) h(t, M) = e Mt + e M(T t)

13 The PCAC mass and the chiral limit (a m) (1/κ) m eff (t) = Cγ 0γ 5,γ 5 (t + 1) C γ0 γ 5,γ 5 (t 1) C γ5,γ 5 (t) 1 am = A κ 1 «κ c = (7) κ c

14 The PS mass 7 6 a M 2 PS /m am a 2 M 2 PS = Bam

15 The V mass am V am M V = M V (0) + Dm

16 The PS decay constant af PS am C γ5,γ 5 (t) G2 PS exp{ M PS t} M PS F PS = m MPS 2 G PS F PS = F PS (0) + Cm

17 Validity region of χpt 0.25 M PS /(4 π F PS ) am M PS 4πF PS 0.2 We are at the superior corner of the region of applicability of χpt. So far data are compatible with the standard scenario of chiral symmetry breaking at the chiral point. Anyway more exotic scenarios (like the presence of a conformal chiral point) cannot be excluded. We need to go closer to the chiral point in a safe way (increasing the volume and reducing the lattice spacing).

18 Outline 1 Motivations Why higher representations? 2 The HiRep code The HiRep code 3 SU(2) with n f = 2 adjoint Conformal point? The parameters The chiral limit Troubles at small m at fixed lattice spacing Extracting the masses from correlators Results 4 Conclusions

19 Conclusions SU(N) gauge theories with fermions in two-index representations are relevant for the physics beyond SM. We have implemented and tested the HMC/RHMC algorithm for fermions in the generic representation of SU(N). We have produced some preliminary phenomenological results for SU(2) with n f = 2 adjoint fermions at fixed lattice spacing. Our results are affected by systematic errors, due to both finite lattice spacing and finite volume. In particular the chiral limit and the scaling region require deeper investigation. Our results are compatible with the standard scenario of chiral symmetry breaking in the chiral point and χpt. However more exotic scenarios cannot be excluded. Lighest quarks are necessary.

20 Behaviour of the HMC/RHMC algorithm If H is the difference of the value of the Hamiltonian at the beginning and at the end of the MD evolution, we expect If t is the MD step size, we expect exp( H) = 1 H t 4 If P acc is the acceptance probability, we expect P acc = erfc( p H /2) The average of the plaquette is independent of t. Violation of reversibility. Fix a starting configuration, evolve for τ = 1, flip the momenta and evolve back for τ = 1. The starting and ending configurations should be the same. We get δh 10 7

21 Test of the group structure SU(3) + n f = 2 in the fundamental representation, checked against: M. Luscher, Comput. Phys. Commun. 165, 199 (2005) [arxiv:hep-lat/ ] SU(3) + n f = 2 in the fundamental representation = = SU(3) + n f = 2 in the antisymmetric two-index representation SU(2) + n f = 2 in the adjoint representation, checked against: S. Catteral and F. Sannino, Phys. Rev. D 76, (2007) [arxiv:hep-lat/ ] SU(2) + n f = 2 in the adjoint representation = = SU(2) + n f = 2 in the symmetric two-index representation

22 The lowest eigenvalue of D κ= κ= κ= κ= κ= aλ

23 Failure of the naive TC scaling lattice 4 3 lattice m V / F PS am From the naive TC scaling: M V F PS = Mρ F π 8.4

24 The PS mass ours Catteral et al. 1 am PS /k

25 The V mass ours Catteral et al am V /k

26 Compatibility with exotic scenarios ours Catterall et al. M V = b M PS am V = c (am PS ) 2 am V am PS