Probing triple-gauge couplings in anomalous gauge theories at hadron and lepton colliders

Gauge anomalous quantum field theories are inconsistent as full UV theories since they lead to the breaking of Lorentz invariance or unitarity, as well as nonrenormalizability. It is well known, however, that they can be interpreted as effective field theories (EFT) with a cutoff. The latter cannot be made arbitrarily large, and it is related to the energy scale at which additional fermions with suitable gauge charges enter, rendering the full model anomaly free. A nondecoupling effect that remains in the EFT is the appearance of anomalous loop-induced triple-gauge couplings, encapsulating information from the full UV theory. In this work, we take as an example an Abelian gauge symmetry <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:msubsup><a:mo stretchy="false">)</a:mo><a:mi>μ</a:mi><a:mo>′</a:mo></a:msubsup></a:math> under which second-generation leptons are axially charged, leading to an EFT that consists of the Standard Model (SM) with an additional massive <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>Z</e:mi><e:mo>′</e:mo></e:msup></e:math> gauge boson. As a consequence, there are triple-gauge couplings involving the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>Z</g:mi><g:mo>′</g:mo></g:msup></g:math> and electroweak SM gauge bosons via mixed gauge anomalies. We study the possibility of probing these loop suppressed anomalous couplings at hadron and lepton colliders, with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:msup><i:mrow><i:mi>Z</i:mi></i:mrow><i:mrow><i:mo>′</i:mo></i:mrow></i:msup></i:mrow></i:math>-lepton couplings allowed by current experimental bounds, finding that due to the large SM backgrounds and small signal, the HL-LHC is incapable of this task. The 100 TeV <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>p</k:mi><k:mi>p</k:mi></k:math> collider at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi mathvariant="script">L</m:mi><m:mo>=</m:mo><m:mn>20</m:mn><m:mtext> </m:mtext><m:mtext> </m:mtext><m:msup><m:mi>ab</m:mi><m:mrow><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:math> on the other hand could probe anomalous couplings for <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mrow><p:msub><p:mrow><p:mi>m</p:mi></p:mrow><p:mrow><p:msup><p:mrow><p:mi>Z</p:mi></p:mrow><p:mrow><p:mo>′</p:mo></p:mrow></p:msup></p:mrow></p:msub><p:mo>∈</p:mo><p:mo stretchy="false">[</p:mo><p:mn>150</p:mn><p:mo>,</p:mo><p:mn>800</p:mn><p:mo stretchy="false">]</p:mo><p:mtext> </p:mtext><p:mtext> </p:mtext><p:mi>GeV</p:mi></p:mrow></p:math> and obtain discovery significances for <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msub><t:mi>m</t:mi><t:msup><t:mi>Z</t:mi><t:mo>′</t:mo></t:msup></t:msub><t:mo>∈</t:mo><t:mo stretchy="false">[</t:mo><t:mn>230</t:mn><t:mo>,</t:mo><t:mn>330</t:mn><t:mo stretchy="false">]</t:mo><t:mtext> </t:mtext><t:mtext> </t:mtext><t:mi>GeV</t:mi></t:math>. Lepton colliders are also well suited for probing these anomalous couplings. In particular, we show that a muon collider running at the <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msup><x:mi>Z</x:mi><x:mo>′</x:mo></x:msup></x:math> resonance and an electron-positron collider such as CLIC with <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msqrt><z:mi>s</z:mi></z:msqrt><z:mo>=</z:mo><z:mn>3</z:mn><z:mtext> </z:mtext><z:mtext> </z:mtext><z:mi>TeV</z:mi></z:math> can be complimentary in probing the anomalous couplings for <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msub><bb:mi>m</bb:mi><bb:msup><bb:mi>Z</bb:mi><bb:mo>′</bb:mo></bb:msup></bb:msub><bb:mo>∈</bb:mo><bb:mo stretchy="false">[</bb:mo><bb:mn>100</bb:mn><bb:mo>,</bb:mo><bb:mn>700</bb:mn><bb:mo stretchy="false">]</bb:mo><bb:mtext> </bb:mtext><bb:mtext> </bb:mtext><bb:mi>GeV</bb:mi></bb:math>, with CLIC sensitive to discovery for <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mrow><fb:msub><fb:mrow><fb:mi>m</fb:mi></fb:mrow><fb:mrow><fb:msup><fb:mrow><fb:mi>Z</fb:mi></fb:mrow><fb:mrow><fb:mo>′</fb:mo></fb:mrow></fb:msup></fb:mrow></fb:msub><fb:mo>∈</fb:mo><fb:mo stretchy="false">[</fb:mo><fb:mn>125</fb:mn><fb:mo>,</fb:mo><fb:mn>225</fb:mn><fb:mo stretchy="false">]</fb:mo><fb:mtext> </fb:mtext><fb:mtext> </fb:mtext><fb:mi>GeV</fb:mi></fb:mrow></fb:math>.