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On the spectrum of light mesons in an SU(2) gauge theory with - - PowerPoint PPT Presentation

Aug 14, 2022 206 likes •425 views

On the spectrum of light mesons in an SU(2) gauge theory with dynamical fermions Reinhard Alkofer 1 and Milan Vujinovic 2 1 Institute of Physics, University of Graz 2 Sao Carlos Institute of Physics, University of Sao Paulo 8th International

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SLIDE 1

On the spectrum of light mesons in an SU(2) gauge theory with dynamical fermions

Reinhard Alkofer1 and Milan Vujinovic2

1Institute of Physics, University of Graz 2Sao Carlos Institute of Physics, University of Sao Paulo

8th International Conference on the Exact Renormalization Group, Trieste, Sept. 19, 2016

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 1 / 18

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SLIDE 2

Outline

1

Motivation: A Minimal Template for Extensions of the SM

2

Fermion—gauge-boson vertex

3

Numerical results for bound state masses

4

Conclusions and Outlook

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 2 / 18

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SLIDE 3

Motivation: A Minimal Test Case for Model Building

SU(2) gauge theory with two fundamentally charged Dirac fermions simplest field theoretical realization of unified theory of Composite Goldstone Boson Higgs and Technicolor bosonic mesons and “baryons“ (Pauli-Gürsey symmetry) dynamically broken chiral symmetry SU(4) → SP(4) ∼ SO(5): five Goldstone bosons in chiral limit breaking direction w.r.t. SM:

vanishing angle: Composite Goldstone Boson Higgs four GB: (massless) complex Higgs doublet; 5th: SM neutral maximal angle: Technicolor Theory EW symmetry completely broken, 3 GB → long. W/Z states, Higgs = lightest scalar meson, 2 GB: dark matter candidates

angle determined dynamically: GB Higgs mixes with technicolor scalar meson ⇒ two scalars with lighter one = Higgs1

1Back-coupling to SM particles will lead to large corrections!

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 3 / 18

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SLIDE 4

Motivation

Spectrum of this theory has been calculated using lattice simulations [R. Arthur et al. [CP3], arXiv:1602.06559 [hep-lat]; arXiv:1607.06654 [hep-lat];

and references therein]

with main conclusion: Significantly different spectrum than in Nf = 2 SU(3) gauge theory (i.e. QCD)! Try to understand the spectra by using another non-perturb. approach!

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 4 / 18

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SLIDE 5

Motivation

Functional (continuum) methods include: Dyson-Schwinger — Bethe-Salpeter — cov. Faddeev eqs. nPI effective actions scale-dependent bosonisation in ERG

see poster by Jordi Paris Lopez

Suitable for multi-scale problems! (Proximity of conformal window) Bound-state formation in terms of microscopic d.o.f.! Truncation is necessary

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 5 / 18

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SLIDE 6

Fermion—gauge-boson vertex

Truncation? Refer to symmetries expressed via Ward ids. and/or nPI actions . . . Example: 3PI → Kernel of the fermion-antifermion Bethe-Salpeter eq.

= + + (..)

R.A., C.S. Fischer, F.J. Lllanes-Estrada, Mod.Phys.Lett. A23 (2008) 1105; H.Sanchis-Alepuz, R.Williams, J.Phys.Conf.Ser.631 (2015)012064 [arXiv:1503.05896]

Fully dressed fermion—gauge-boson vertices!

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 6 / 18

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SLIDE 7

Fermion—gauge-boson vertex

Kernel of ψ ¯ ψ BS equation via 3PI action:

= + + (..)

Needed input: Gauge-boson propagator and fermion—gauge-boson vertex!

  • Eq. from 3PI action:

= − 1

2Nc

+Nc

2

Needed input: Fermion & gauge-boson propagator and 3-gauge-boson vertex! These 2- & 3-point functions are needed for complex arguments to describe bound states with physical masses (timelike momentum)!

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 7 / 18

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SLIDE 8

Fermion—gauge-boson vertex

Relative importance of the eight transverse tensor structures: dynamically generated χSB coupling important! Technically required truncation of the fermion–gauge boson vertex has to be chosen very careful!

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 8 / 18

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SLIDE 9

Fermion—gauge-boson vertex

Truncation I: 1PI-based

= + +

Non−Abelian Abelian Tree

but not fully self-consistent

= + +

Non−Abelian Abelian Tree

Truncation II: “3PI“, partially self-consistent, simplified 3-gauge-boson vertex

Tree Ghost triangle

= +

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 9 / 18

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SLIDE 10

Bound state equation

=

+ +

K

+ + +

In truncation II (i.e., 3PI-type): Additional dressings for the 3-gauge-boson and the fermion–gauge-boson vertices.

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 10 / 18

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SLIDE 11

Numerical results for input

Gauge-boson and ghost renormalization function

10-3 10-2 10-1 100 101

10-3 10-2 10-1 100 101 102 103 104

p2 [arb. units2]

Z(p2) G(p2)

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 11 / 18

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SLIDE 12

Numerical results for input

3-gauge-boson vertex as function of s0 = 1

6(p2 1 + p2 2 + p2 3):

  • 2
  • 1

1 2 3 10-4 10-3 10-2 10-1 100 101 102 103 104

Γ3g dressing

s0 [arb. units2]

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 12 / 18

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SLIDE 13

Numerical results for input

Quark mass function

0.05 0.1 0.15 0.2 0.25 0.3

10-4 10-3 10-2 10-1 100 101 102 103 104

p2 [arb. units2]

Mass [arb. units]

NA 1PI NA + AB 1PI NA 3PI NA + AB 3PI

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 13 / 18

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SLIDE 14

Numerical results for input

Ledaing tensor structure fermion–gauge-boson vertex

0.5 1 1.5 2 2.5 3 3.5 4 10-4 10-3 10-2 10-1 100 101 102 103 104

p2 [arb. units2]

NA 1PI NA + AB 1PI NA 3PI NA + AB 3PI

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 14 / 18

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SLIDE 15

Numerical results for input

Positivity violation for quark propagator

10-5 10-4 10-3 10-2 10-1 100

10 20 30 40 50 |σS(t)|

t [1/arb. unit]

1PI approach 3PI approach

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 15 / 18

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SLIDE 16

Numerical results for bound state masses

Scale set by fPS = 0.246 TeV JPC NA, 1PI NA + AB, 1PI NA, “3PI “ NA + AB, “3PI“ Lattice 0−+ – 0++ 1.39(3) 1.22(2) 1.33(3) 1.25(2) 4.7 (2.7) 1−− 2.27(5) 2.00(4) 2.37(5) 1.99(4) 3.2(5) 1++ 2.87(5) 2.65(5) 3.09(6) 2.67(5) 3.6(9)

Lattice:

  • R. Arthur et al. [CP3], arXiv:1602.06559 [hep-lat]; arXiv:1607.06654 [hep-lat];

and references therein

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 16 / 18

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SLIDE 17

Numerical results for bound state masses

Light scalar meson in (unquenched) QCD from the 3PI effective action: RL 2PI - 3L 3PI - 3L m0++/fπ 6.96 5.05 10.5 ± 1.0 m1−−/fπ m1++/fπ 3PI-3L 7.0 12.4 ± 1.0

  • R. Williams, C.S. Fischer, W. Heupel, Phys.Rev. D93 (2016) 034026

Light scalar meson in SU(2) gauge theory with two light flavours: 1PI 3PI-type m0++/fPS † 5.0 ± 0.1 5.1 ± 0.1 m1−−/fπ m1++/fπ 3PI-type 8.1 ± 0.2 10.9 ± 0.2

R.A., M. Vujinovic, to be published

† with non-Abelian diagram only: 5.7 ± 0.1 and 5.4 ± 0.1

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 17 / 18

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SLIDE 18

Conclusions and Outlook

QCD(-like) gauge theories in Landau gauge:

Established results for propagators & recent investigations of three-point functions

No reliable quantitative results for bound states without

self-consistent fermion–gauge-boson vertex!

Unexpected results for SU(2) bound state spectrum . . .

Significantly lower masses than corresponding lattice values! Significantly different from SU(3) / QCD ?!

Approach merging different functional methods

to be applied soon to different gauge theories / matter content: Dynamical electroweak symmetry breaking & strongly-interacting dark matter.

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 18 / 18

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SLIDE 19

Conclusions and Outlook

QCD(-like) gauge theories in Landau gauge:

Established results for propagators & recent investigations of three-point functions

No reliable quantitative results for bound states without

self-consistent fermion–gauge-boson vertex!

Unexpected results for SU(2) bound state spectrum . . .

Significantly lower masses than corresponding lattice values! Significantly different from SU(3) / QCD ?!

Approach merging different functional methods

to be applied soon to different gauge theories / matter content: Dynamical electroweak symmetry breaking & strongly-interacting dark matter.

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 18 / 18

slide-20
SLIDE 20

Conclusions and Outlook

QCD(-like) gauge theories in Landau gauge:

Established results for propagators & recent investigations of three-point functions

No reliable quantitative results for bound states without

self-consistent fermion–gauge-boson vertex!

Unexpected results for SU(2) bound state spectrum . . .

Significantly lower masses than corresponding lattice values! Significantly different from SU(3) / QCD ?!

Approach merging different functional methods

to be applied soon to different gauge theories / matter content: Dynamical electroweak symmetry breaking & strongly-interacting dark matter.

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 18 / 18

slide-21
SLIDE 21

Conclusions and Outlook

QCD(-like) gauge theories in Landau gauge:

Established results for propagators & recent investigations of three-point functions

No reliable quantitative results for bound states without

self-consistent fermion–gauge-boson vertex!

Unexpected results for SU(2) bound state spectrum . . .

Significantly lower masses than corresponding lattice values! Significantly different from SU(3) / QCD ?!

Approach merging different functional methods

to be applied soon to different gauge theories / matter content: Dynamical electroweak symmetry breaking & strongly-interacting dark matter.

  • R. Alkofer (Theoretical Physics, U. Graz)

Meson spectrum in SU(2) gauge theory ERG 2016, Sept. 19, 2016 18 / 18