TQFT2023@KEK, Aug 28

New gauge-independent transition separating
confinement-Higgs phase in the latticegauge-fundamental scalar model
Ryu Ikeda (Chiba University)
Collaborate with:
Kei-Ichi Kondo (Chiba University)
Akihiro Shibata (KEK)
Seikou Kato (Oyama National College of Technology)
arXiv:2308.13430 [hep-lat]1

1. Introductionn We consider SU(2) LGST (with fundamental scalar)
Action of SU(2) LGST (with fund. scalar):・・: link variables,: gauge coupling,: (normalized) scalar fields: scalar couplingtransforms as the fundamental representation of the gauge group SU(2).
This model has thewheresymmetry:,.
2n Motivationp

In case of fundamental scalar field → This talkConfinement () and Higgs ()regions are subregions of an analytically continued single
phase. The transition line starts fromdoes notreach “analytic region” which connect these subregions.
[1] K. Osterwalder and E. Seiler, Annals Phys. 110, 440 (1978)[2] E. Fradkin and S. H. Shenker, Phys. Rev. D19, 3682 (1979)

pIn case of adjoint scalar field → [3]Confinement and Higgs regions are completely separated intothe two different phases by the continuous transition line.
The transition line has two endpoints,

and.
[3] A. Shibata and K.-I. Kondo (2023), arXiv: 2307.15953 [hep-lat][4] R. C. Brower et al., Phys. Rev. D25, 3319 (1982)
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n Motivation and Results
Re-examine the 4D SU(2) lattice gauge scalar theory (LGST) with fund. scalar field,pWe found the gauge-invariant composite operator of the original scalar field andthe new “color-direction field”, which enables to separate the confinement phaseand the Higgs phase completely and gauge-independently.
(Greensite-Matsuyama discriminated 2 phases by detecting the SSB of)[5] J. Greensite and K. Matsuyama, Phys. Rev. D98, 074504 (2018); Phys. Rev. D101, 054508 (2020)p

We perform the numerical simulations for this model without any gauge fixing, andfound a new transition line:
・ in the weak gauge coupling, it agrees with the conventional thermodynamicaltransition line.
・ in the strong gauge coupling, it divides the single confinement-Higgs phase into
two separate phases, confinement and the Higgs.
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2. New transition line and Color-direction fieldn Newly found transition linep

As the result of the numerical simulations for the4D SU(2) LGST with fund. scalar, we found a newtransition line which separates confinement andHiggs regions completely, without any gauge fixing.

pThis transition line is obtained in the gaugeindependent way by introducing the new composite
operator of the original scalar field and the new“color-direction field”, based on the gauge-covariant
decomposition of the gauge field due to [6-9].
[6] Y.M. Cho, Phys. Rev. D21, 1080 (1980); Phys. Rev. D23, 2415 (1981)[7] Y.S. Duan and M.L. Ge, Sinica Sci. 11, 1072 (1979)[8] S.V. Shabanov, Phys. Lett. B463, 263 (1999)[9] L.D. Faddeev and A.J. Niemi, Phys. Rev. Lett. 82, 1624 (1999); Nucl. Phys. B776, 38 (2007)5

n gauge-covariant decomposition of the gauge field
is gauge-covariantly decomposable:(We required the transformations,)

・This decomposition is given uniquely by solving the defining equations for the“color-direction field”

with a unit :(We required the transformation,
・For a given set of gauge fields), a set of color-direction fields

is determinedas the configuration minimizing the reduction functional:・has the same form as the Higgs action of SU(2) LGST (with adj. scalar)(minimization ofextracts the DOF offrom the gauge fields)[10] Kondo et al., Phys. Rep. 579, 1-226 (2015)6!
n construction of the scalar-color density 𝑹
・ We required that

whereand,transform as.

・ We can define a global covariant local field

and its spacetime average:

・ Then there exists a non-trivial gauge invariant defined asalso can be defined as the absolute value of the two eigenvalues of.
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3. Numerical simulationn Setting for lattice simulationplattice, pseudo heat bath methodpcold start ()p

5000 sweeps for thermalization before samplingpa gauge/scalar config. sampling per 100 sweeps(100 samples for average)

To determine a set ofp

reduction functionalfrom a given set of,is minimized by theiterative method with over-relaxation.
p

10 trials are taken for searching global optimal config..

Performed for 17×52=884 sets of parameter points.

Transition lines were identified from gaps/rises of the plots.
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n plaquette density 𝑃

9n scalar density 𝑀

10n Transition lines determined by 𝑃 , 𝑀

11n modulus of scalar-color density (3rd component) |𝑅& !| NEW12

n scalar-color density norm!
𝑹NEW

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n Transition lines determined by |𝑅& !| ,!
𝑹NEW

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n Phase structure and physical interpretation(Ⅰ) confinement phase ( 𝑅 ! = 0) …

unbroken phasep conﬁnement similar to the pure SU(2) gauge theory
p Isotropic color-direction ﬁeld config.

in color spacep very small correlation between the color-direction ﬁeld

andthe scalar ﬁeld
p Massive gauge ﬁelds due to self-interactions among themselves(Ⅱ) Higgs phase ( 𝑅 ! > 0) …

broken phasep conventional BEH mechanism with complete SSB SU(2)→1
p Anisotropic color-direction ﬁeld config.

in color spacep stronger correlation between the color-direction ﬁeld

scalar ﬁeldand the(which tends to align to a particular direction)p Massive gauge ﬁelds due to the BEH mechanism
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4. Summary
We studied the new type of operator in the SU(2) LGST with fundamental scalar field.
n Combining the scalar fieldand newly introduced “color-direction field”(representing the gauge DOF) as the composite operator, we found the complete andgauge-independent separation between the confinement phase and the Higgs phase.
n We performed the gauge-fixing-free numerical simulations and checked that there is a
new transition line which overlaps with the known thermodynamical transition line in the
weak gauge coupling, and divides a single confinement-Higgs phase into the two phasesin the strong gauge coupling.
n New transition lines can be interpretated as the consequence of the SSB of.

Outlook:Ø More physical meaning and implications of the new transition(e.g. quark-hadron continuity, Schäfer and Wilczek, 1999, …)
Ø Improvement of the accuracy for the numerical simulation
Ø Extension to the case of LGST with SU(N) gauge group (𝑁 ≥ 3)
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5. Discussionn Consistency with the OSFS theorem
p In Osterwalder-Seiler [1], the lattice action is written in terms of the scalar ﬁeld in

the SU(2) doublet, therefore the custodial symmetry does not assumed.
p In Fradkin-Shenker [2], the lattice action agrees with ours, however in estimating theanalytic region, the unitary gauge

breaks the custodial symmetry.

→ New transition line does not contradict with the OSFS theorem.

n Absence of the massless Nambu-Goldstone particles
p In the Higgs phase, the continuous global symmetry is spontaneously broken.
p The color-direction ﬁeld

obtained by minimizing the global reduction functional

is intrinsically non-local, which violates one of the assumptions of the NG theorem.
→ There are no massless particles (gapless excitations) in the Higgs phase, although
the continuous global symmetry is spontaneously broken.
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