Curvature-Restored Gauge Invariance and Ultraviolet Naturalness

It is shown that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mo stretchy="false">(</mml:mo><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>H</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mi>R</mml:mi></mml:math> in a spacetime of curvature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math> is a natural ultraviolet <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">U</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> completion of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mo stretchy="false">(</mml:mo><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">|</mml:mo><mml:mi>H</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> in the flat-spacetime Standard Model <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with Higgs field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi mathvariant="normal">U</mml:mi><mml:mi mathvariant="normal">V</mml:mi></mml:math> scale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math>, and loop factors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:math>. This curvature completion rests on the fact that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math>-mass gauge theory in flat spacetime turns, on the cut view <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">4</mml:mn><mml:msup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup></mml:math>, into a massless gauge theory in curved spacetime. It provides a symmetry reason for curved spacetime, wherein gravity and matter are both low-energy effective phenomena. Gravity arises correctly if new physics exists with at least 63 more bosons than fermions, with no need to interact with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">M</mml:mi></mml:math> and with dark matter as a natural harbinger. It can source various cosmological, astrophysical, and collider phenomena depending on its spectrum and couplings to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">M</mml:mi></mml:math>.

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