$\boldsymbol{C\!P}\!$ violation in Higgs-gauge interactions: from tabletop experiments to the LHC

TL;DR

This paper explores how $CP$-violating Higgs-gauge interactions affect low- and high-energy experiments, showing low-energy measurements currently provide stronger constraints than the LHC, with implications for future collider searches.

Contribution

It systematically analyzes $CP$ violation in Higgs-gauge interactions using effective field theory, connecting low-energy EDM and $B$ decay constraints with high-energy collider prospects.

Findings

Low-energy measurements impose stronger constraints than the LHC.

Allowed regions in coupling space are highly correlated and distinctive.

High-luminosity LHC can probe the characteristic patterns of $CP$ violation.

Abstract

We investigate the interplay between the high- and low-energy phenomenology of $C\!P$-violating interactions of the Higgs boson with gauge bosons. For this purpose we use an effective field theory approach and consider all dimension-6 operators arising in so-called universal theories. We compute their loop-induced contributions to electric dipole moments and the $C\!P$ asymmetry in $B\to X_s\gamma$, and compare the resulting current and prospective constraints to the projected sensitivity of the LHC. Low-energy measurements are shown to generally have a far stronger constraining power, which results in highly correlated allowed regions in coupling space, a distinctive pattern that could be probed at the high-luminosity LHC.

Tables

Table 1. Table 1: Current limits on the electron  Andreev:2018ayy , neutron  Baker:2006ts ; Afach:2015sja , mercury  Griffith:2009zz ; Graner:2016ses , xenon  PhysRevLett.86.22 ; Sachdeva:2019rkt , and radium  Parker:2015yka ; Bishof:2016uqx EDMs in units of e 𝑒 e cm ( 90 % percent 90 90\% C.L.). The result for the C ​ P 𝐶 𝑃 C\!P\! asymmetry, A B → X s ​ γ = 0.015 ​ ( 20 ) subscript 𝐴 → 𝐵 subscript 𝑋 𝑠 𝛾 0.015 20 A_{B\to X_{s}\gamma}=0.015(20) , is taken from Refs.  Nishida:2003paa ; Lees:2014uoa ; Amhis:2016xyh .

$d_e$	$d_n$	$d_{\mathrm{Hg}}$	$d_{\mathrm{Xe}}$	$d_{\mathrm{Ra}}$
$1.1\cdot {10}^{-29}$	$3.0\cdot {10}^{-26}$	$6.2\cdot {10}^{-30}$	$3.9\cdot {10}^{-27}$	$1.2\cdot {10}^{-23}$

Table 2. Table 2: Central and Rfit low-energy constraints (at 95 % percent 95 95\% C.L.), assuming one of the couplings, C α subscript 𝐶 𝛼 C_{\alpha} , is present at the scale Λ = 1 Λ 1 \Lambda=1 TeV. For comparison, we show current collider limits from Refs.  Sirunyan:2019twz ; Sirunyan:2019nbs (for C φ ​ B ~ subscript 𝐶 𝜑 ~ 𝐵 C_{\varphi\tilde{B}} ), Refs.  Aaboud:2017fye ; Aaboud:2019nkz (for C φ ​ W ~ ​ B subscript 𝐶 𝜑 ~ 𝑊 𝐵 C_{\varphi\tilde{W}B} and C W ~ subscript 𝐶 ~ 𝑊 C_{\tilde{W}} ), and Ref.  Bernlochner:2018opw (for all other couplings).

	Central	Rfit	LHC
$v^2{C}_{\phi \tilde{B}}$	$[-5.1,5.1]\cdot {10}^{-6}$	$[-5.1,5.1]\cdot {10}^{-6}$	$[-28,10]$
$v^2{C}_{\phi \tilde{W}}$	$[-4.7,4.7]\cdot {10}^{-6}$	$[-4.7,4.7]\cdot {10}^{-6}$	$[-2.3,0.43]$
$v^2{C}_{\phi \tilde{W}B}$	$[-2.2,2.2]\cdot {10}^{-6}$	$[-2.2,2.2]\cdot {10}^{-6}$	$[-0.57,0.57]$
$v^2{C}_{\phi \tilde{G}}$	$[-5.3,5.3]\cdot {10}^{-5}$	$[-1.2,1.2]\cdot {10}^{-3}$	$[-1.3,8.1]\cdot {10}^{-3}$
$v^2{C}_{\tilde{G}}$	$[-2.4,2.4]\cdot {10}^{-6}$	$[-3.4,3.4]\cdot {10}^{-5}$	–
$v^2{C}_{\tilde{W}}$	$[-4.8,4.8]\cdot {10}^{-5}$	$[-4.8,4.8]\cdot {10}^{-5}$	$[-3.1,3.1]\cdot {10}^{-2}$

Table 3. Table 3: Comparison of projected collider and low-energy limits. The LHC limits were taken from Ref.  Bernlochner:2018opw , while the low-energy limits assume improved matrix elements and future EDM measurements as described in the text. All four couplings were turned on at the scale Λ = 1 Λ 1 \Lambda=1 TeV, and the low-energy limits were obtained using the Gaussian strategy for the theoretical uncertainties.

	Low energy	LHC (3000 fb-1)
$v^2{C}_{\phi \tilde{B}}$	$[-0.4,0.00]$	$[-0.3,0.3]$
$v^2{C}_{\phi \tilde{W}}$	$[-2.3,0.02]$	$[-0.17,0.17]$
$v^2{C}_{\phi \tilde{W}B}$	$[-1.3,0.01]$	$[-0.39,0.39]$
$v^2{C}_{\phi \tilde{G}}$	$[-1.3,1.3]\cdot {10}^{-5}$	$[-9.0,9.0]\cdot {10}^{-4}$

Table 4. Table 4: Coefficients of the fermion EDMs, quark CEDMs, and Weinberg operator at μ = 2 𝜇 2 \mu=2 GeV in units of 10 − 2 superscript 10 2 10^{-2} , assuming μ 0 = Λ = 1 subscript 𝜇 0 Λ 1 \mu_{0}=\Lambda=1 TeV. The label ( d ) 𝑑 (d) denotes both the operators involving down and strange quarks, while the superscript q 𝑞 q in c ~ g ( q ) subscript superscript ~ 𝑐 𝑞 𝑔 \tilde{c}^{(q)}_{g} denotes q = { u , d , s } 𝑞 𝑢 𝑑 𝑠 q=\{u,d,s\} .

	$C_{\phi \tilde{B}}$	$C_{\phi \tilde{W}}$	$C_{\phi \tilde{W}B}$	$C_{\phi \tilde{G}}$	$C_{\tilde{G}}$	$C_{\tilde{W}}$
${\tilde{c}}_{\gamma }^{\left(e\right)}$	$-1.4$	$-1.5$	$3.3$	–	–	$0.14$
${\tilde{c}}_{\gamma }^{\left(u\right)}$	$-0.62$	$-1.1$	$2.3$	$6.5$	$-6.2$	$0.11$
${\tilde{c}}_{\gamma }^{\left(d\right)}$	$-0.31$	$-2.2$	$4.0$	$6.5$	$-6.2$	$0.22$
${\tilde{c}}_g^{\left(q\right)}$	–	–	–	$10$	$-15$	–
$C_{\tilde{G}}$	–	–	–	$-0.22$	$23$	–
$C_7$	–	–	$-9.9v^2$	–	–	$1.7v^2$

Equations

$$\mathcal{L}$$

$$-g{g}^{\prime }{C}_{\phi \tilde{W}B}{\phi }^{†}{\tau }^i\phi {\tilde{W}}_{\mu \nu }^i{B}^{\mu \nu }-g_s^2{C}_{\phi \tilde{G}}{\phi }^{†}\phi G_{\mu \nu }^a{\tilde{G}}_a^{\mu \nu }$$

$$+\frac{C_{\tilde{G}}}{3}{g}_s{f}_{abc}{\tilde{G}}_{\mu \nu }^a{G}_b^{\nu \rho }{G}_{\rho }^{c\mu }+\frac{C_{\tilde{W}}}{3}g{ϵ}_{ijk}{\tilde{W}}_{\mu \nu }^i{W}_j^{\nu \rho }{W}_{\rho }^{k\mu },$$

$$v^2{C}_{\phi \tilde{W}B}=-0.93_{-0.31}^{+0.47},v^2{C}_{\tilde{W}}=0.42(33).$$

$${\mathcal{L}}_{\mathrm{EDMs}}$$

$$+\sum _{f=u,d,s,c,b,t}\left(C_g^{(f)}{O}_g^{(f)}+\text{h.c.}\right)+C_{\tilde{G}}{O}_{\tilde{G}},$$

$${\mathcal{L}}_{b\to s}$$

$$O_g^{(f)}$$

$$O_{\gamma }^{(f)}$$

$$O_7$$

$${\tilde{c}}_{\gamma }^{(f)}$$

$$\displaystyle-\frac{2T^{3}_{f}}{s_{w}^{2}Q_{f}}C_{\varphi\tilde{W}}+\frac{3T^{3}_{f}+t_{w}^{2}(2Q_{f}-T^{3}_{f})}{s_{w}^{2}Q_{f}}C_{\varphi\tilde{W}B}\bigg{)}$$

$$-2\frac{T_f^3-2s_w^2{Q}_f}{s_w^2{Q}_f}\frac{m_Z^2}{m_h^2-m_Z^2}\log \frac{m_h^2}{m_Z^2}$$

$$\times \left(t_w^2{C}_{\phi \tilde{B}}-C_{\phi \tilde{W}}+\frac{1-t_w^2}{2}{C}_{\phi \tilde{W}B}\right)$$

$$\displaystyle-\frac{T^{3}_{f}}{s_{w}^{2}Q_{f}}\left(2\log\frac{m_{W}^{2}}{m_{h}^{2}}C_{\varphi\tilde{W}B}-C_{\tilde{W}}\right)\bigg{]}\,,$$

$${\tilde{c}}_g^{(q)}$$

$$C_{\tilde{G}}({\mu }_t^{-})$$

$$C_7({\mu }_t)$$

$$f(x)$$

$$g(x)$$

$${\omega }_{\text{ThO}}=120.6(4.9)\mathrm{mrad}/\mathrm{s}\left(\frac{d_e}{10^{-27}e\mathrm{cm}}\right),$$

$$d_n$$

$$-0.0028(17)d_s-0.55(28)e{\tilde{d}}_u$$

$$-1.10(55)e{\tilde{d}}_d+50(40)\mathrm{MeV}e{g}_s{C}_{\tilde{G}},$$

$$d_p$$

$$-0.0028(17)d_s+1.30(65)e{\tilde{d}}_u$$

$$+0.6(3)e{\tilde{d}}_d-50(40)\mathrm{MeV}e{g}_s{C}_{\tilde{G}},$$

$$\mathcal{L}_{\pi N}=\bar{g}_{0}\bar{N}\mbox{\boldmath$\tau$}\cdot\mbox{\boldmath$\pi$}N+\bar{g}_{1}\bar{N}\pi_{3}N\,,$$

$${\bar{g}}_0$$

$${\bar{g}}_1$$

$$d_D=0.94(1)(d_n+d_p)+0.18(2){\bar{g}}_{1}e\mathrm{fm}.$$

$$d_{\mathrm{Hg}}$$

$$\displaystyle+\bigg{(}0.13^{+0.50}_{-0.07}\,\bar{g}_{0}+0.25^{+0.89}_{-0.63}\,\bar{g}_{1}\bigg{)}e\,{\rm fm}\bigg{]}$$

$$+0.012(12)d_e,$$

$$d_{\mathrm{Ra}}=-7.7(8)\cdot 10^{-4}[-2.5(7.6){\bar{g}}_0+63(38){\bar{g}}_1]e\mathrm{fm}.$$

$$d_{\mathrm{Xe}}$$

$$\displaystyle+\left(-0.10^{+0.04}_{-0.53}\,\bar{g}_{0}-0.08^{+0.04}_{-0.55}\,\bar{g}_{1}\right)e\,{\rm fm}\,\Big{]}\,.$$

$$\frac{A_{CP}}{\pi }$$

$$\left[\left(\frac{40}{81}-\frac{40}{9}\frac{{\Lambda }_c}{m_b}\right)\frac{{\alpha }_s}{\pi }+\frac{{\Lambda }_{17}^c}{m_b}\right]\mathrm{Im}\frac{C_2}{C_7^{\mathrm{tot}}}$$

$$-\left(\frac{4{\alpha }_s}{9\pi }+4\pi {\alpha }_s\frac{{\Lambda }_{78}}{3m_b}\right)\mathrm{Im}\frac{C_8}{C_7^{\mathrm{tot}}}$$

$$-\left(\frac{{\Lambda }_{17}^u-{\Lambda }_{17}^c}{m_b}+\frac{40}{9}\frac{{\Lambda }_c}{m_b}\frac{{\alpha }_s}{\pi }\right)\mathrm{Im}\left({ϵ}_s\frac{C_2}{C_7^{\mathrm{tot}}}\right),$$

$${\Lambda }_{17}^u$$

$${\Lambda }_{17}^c$$

$${\Lambda }_{78}$$

Full text

††preprint: INT-PUB-19-008, LA-UR-19-22027, PSI-PR-19-01, ZU-TH 08/19, RBRC-1316

Introduction.—To generate the observed matter–antimatter asymmetry in the Universe the Sakharov conditions Sakharov:1967dj have to be satisfied. One of them requires that charge-parity ($C\!P$) symmetry be violated. $C\!P\!$ symmetry is broken in the Standard Model (SM) of particle physics with three generations of quarks, but only by the phase of the Cabibbo–Kobayashi–Maskawa (CKM) matrix and, potentially, the QCD $\theta$ term. The resulting amount of $C\!P\!$ violation is, however, far too small to explain the observed matter–antimatter asymmetry Cohen:1993nk ; Gavela:1993ts ; Huet:1994jb ; Gavela:1994ds ; Gavela:1994dt ; Riotto:1999yt . Scenarios of electroweak (EW) baryogenesis Kuzmin:1985mm ; Shaposhnikov:1987tw ; Nelson:1991ab ; Morrissey:2012db demand new sources of $C\!P\!$ violation not too far above the EW scale.

It has long been recognized that the required new $C\!P$-violating couplings can generate observable effects in both Higgs production and decay rates, i.e. $C\!P$-even observables Pospelov:2005pr ; Li:2010ax ; McKeen:2012av ; Harnik:2012pb ; Engel:2013lsa ; Shu:2013uua ; Chen:2014gka ; Inoue:2014nva ; Dwivedi:2015nta ; Chien:2015xha ; Cirigliano:2016njn ; Cirigliano:2016nyn ; Dekens:2018bci , as well as genuinely $C\!P$-odd signatures at the Large Hadron Collider (LHC) Soni:1993jc ; Plehn:2001nj ; Ferreira:2016jea ; Dawson:2013bba ; Anderson:2013afp ; Bernlochner:2018opw ; Alioli:2017ces ; Alioli:2018ljm ; Sirunyan:2019twz ; Sirunyan:2019nbs ; Aaboud:2017fye ; Hirschi:2018etq ; Englert:2019xhk . Moreover, the interplay with low-energy $C\!P$-violating observables such as electric dipole moments (EDMs) has been explored, in either specific models Bian:2014zka ; Inoue:2014nva or SM effective field theory (SMEFT) Pospelov:2005pr ; Li:2010ax ; McKeen:2012av ; Engel:2013lsa ; Inoue:2014nva ; Dwivedi:2015nta ; Chien:2015xha ; Cirigliano:2016njn ; Cirigliano:2016nyn ; Dekens:2018bci , taking into account only subsets of the dimension-6 $C\!P$-odd operators.

Here we take a novel point of view and focus on the $C\!P$-violating sector of so-called universal theories, originally introduced as the broad class of SM extensions in which beyond-the-SM (BSM) particles couple to SM bosons and/or to SM fermions only through the gauge and Yukawa currents Barbieri:2004qk , placing the analysis of the oblique EW corrections Kennedy:1988sn ; Peskin:1990zt and EW precision tests (EWPTs) in a more general and consistent framework.

With the forthcoming high-luminosity (HL) LHC upgrade, EWPTs involving triple-gauge-boson and gauge-boson–Higgs couplings will be an important thrust, and will probe universal theories beyond the level reached with LEP/SLC data Cepeda:2019klc ; Azzi:2019yne ; deBlas:2019rxi . In this context, both for completeness and in connection with baryogenesis, it is timely to study the $C\!P$-violating sector of such theories and to investigate quantitatively the complementarity of collider and low-energy measurements.

To address this problem we work within SMEFT, which relies on assuming a gap between the scale $\Lambda$ of BSM physics and the EW scale. Universal theories induce, modulo field redefinitions, only bosonic operators at the scale $\Lambda$ Wells:2015uba . The SMEFT setup for the $C\!P$-conserving sector of universal theories and the effect of non-universal operators generated by the renormalization group (RG) flow have been studied in Refs. Wells:2015uba ; Wells:2015cre . We find that the $C\!P$-violating sector of universal theories is characterized by six dimension-6 operators, which in the Warsaw basis Buchmuller:1985jz ; Grzadkowski:2010es read

[TABLE]

where $\varphi$ is the Higgs doublet with $\left\langle\varphi\right\rangle=v/\sqrt{2}$, $v\simeq 246\,\text{GeV}$, $g_{s}$, $g$, and $g^{\prime}$ are the $SU(3)_{c}$, $SU(2)_{L}$, and $U(1)_{Y}$ couplings, respectively, and $G_{\mu\nu}$, $W_{\mu\nu}$, and $B_{\mu\nu}$ the corresponding field strength tensors. We define $\tilde{X}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}X_{\alpha\beta}/2$, $\epsilon^{0123}=+1$. The Wilson coefficients $C_{\varphi\,\tilde{X},\tilde{X}}$ encode contributions from BSM physics scaling as $1/\Lambda^{2}$.

This scenario has additional desirable features: it provides a natural arena to study $C\!P$-violating Higgs–gauge interactions in the SMEFT context, as those arise, together with the triple-gauge-boson, as the dominant $C\!P$-violating couplings. Furthermore, the BSM scale $\Lambda$ can be relatively low (as minimal flavor violation DAmbrosio:2002vsn ; Cirigliano:2005ck is satisfied and $C\!P$-violating fermionic dipoles are generated only through RG flow), a welcome feature for the viability of weak-scale baryogenesis.

The operators in Eq. (1) affect the cross sections of processes such as Higgs production via gluon or vector-boson fusion, Higgs production in association with EW gauge bosons, and Higgs decays, through non-interfering contributions quadratic in $C_{\varphi\tilde{X}}$ and are thus suppressed by $(v/\Lambda)^{4}$. Such dimension-8 contributions however still lead to significant constraints Ferreira:2016jea ; Alioli:2018ljm . The Higgs–gauge operators contribute at $\mathcal{O}(v^{2}/\Lambda^{2})$ to $C\!P$-odd observables, such as the $C\!P\!$ asymmetry in $pp\rightarrow h+2j$ Dawson:2013bba ; Anderson:2013afp ; Bernlochner:2018opw ; Sirunyan:2019nbs , angular distributions in associated $HW$ and $HZ$ production Ferreira:2016jea ; Alioli:2017ces ; Alioli:2018ljm , or in $h\rightarrow 4l$ Soni:1993jc ; Sirunyan:2019twz ; Sirunyan:2019nbs , while $C_{\tilde{W}}$ and $C_{\varphi\tilde{W}B}$ contribute to $C\!P$-odd observables in diboson production Ferreira:2016jea ; Aaboud:2017fye . $C_{\tilde{G}}$ gives tree-level corrections to $pp\rightarrow h+2j$ and to multijet production Hirschi:2018etq . In addition to these tree-level effects in collider observables, all coefficients contribute to low-energy $C\!P$-violating observables, such as EDMs and the $C\!P\!$ asymmetry in $B\to X_{s}\gamma$, at the loop level. In this Letter we set up the framework to include low-energy $C\!P$-violating probes and demonstrate that they put severe constraints on the $C\!P$-violating sector of universal theories. To establish the connection to existing collider bounds Bernlochner:2018opw ; Englert:2019xhk , we first concentrate the phenomenological analysis on the operators that involve the Higgs coupling, and later discuss the low- and high-energy input necessary for an analysis of all six parameters simultaneously.

Renormalization group evolution.—When the Higgs field acquires its vacuum expectation value, the operators in Eq. (1) generate $\theta$-like terms by means of $\varphi^{\dagger}\varphi\to v^{2}/2+\dots$, $\varphi^{\dagger}\tau^{i}\varphi\to-\delta^{i3}v^{2}/2+\dots$, where the dots denote terms that contain the Higgs scalar boson $h$. The parts of the operators in Eq. (1) that do not involve $h$ can be absorbed in the SM $\theta$ terms. The $U(1)_{Y}$ and $SU(2)_{L}$ $\theta$ terms are unphysical because they can be removed by field rotations Anselm:1992yz ; Anselm:1993uj ; Perez:2014fja . The gluonic operator effectively shifts the QCD $\theta$ term $\theta\rightarrow\theta-16\pi^{2}v^{2}C_{\varphi\,\tilde{G}}$, which is strongly constrained by the neutron EDM Baker:2006ts ; Afach:2015sja . However, we will assume the presence of a Peccei–Quinn (PQ) mechanism Peccei:1977hh under which the total $\theta$ term vanishes dynamically.

Below the EW scale, the Lagrangian contains flavor-conserving operators that induce leptonic and hadronic EDMs (fermion EDMs, quark chromo EDMs (CEDMs), and the Weinberg operator) as well as $\Delta B=\Delta S=1$ operators that contribute to $B\rightarrow X_{s}\gamma$, through the diagrams shown in Fig. 1. These diagrams provide both finite matching contributions at the EW scale, $\mu=\mu_{t}$, and contributions to the anomalous dimensions that determine the RG evolution between the BSM scale, $\mu=\Lambda$, and the EW scale. We then evolve the low-energy operators to the scale where QCD becomes non-perturbative, $\mu=\Lambda_{\chi}=2$ GeV, and take into account the bottom, charm, and strange thresholds where additional matching contributions are generated. More details about the evolution from the high- to low-energy scale are given in Ref. SuppRG (including Refs. Bobeth:2015zqa ; He:1993hx ; Dekens:2013zca ; Alonso:2013hga ; Chanowitz:1979zu ; tHooft:1972tcz ; Weinberg:1989dx ; Wilczek:1976ry ; Braaten:1990gq ; Degrassi:2005zd ).

A key outcome of the RG analysis is that the weak operators $C_{\varphi\tilde{B}}$, $C_{\varphi\tilde{W}}$, $C_{\varphi\tilde{W}B}$, and $C_{\tilde{W}}$ contribute to the fermion EDMs almost exclusively via two combinations, proportional to the third component of the weak isospin, $T^{3}_{f}$, and the electric charge, $Q_{f}$. For this reason, present and future EDM experiments constrain at most four directions in the parameter space of Eq. (1), up to small subleading effects.

Low-energy observables.—Next, we discuss the connection to the most sensitive low-energy observables, starting with EDMs. The most stringent limits are set by the neutron and 199Hg atom, and by measurements on the polar molecule ThO. For the operators in Eq. (1), the ThO measurement Andreev:2018ayy ; Baron:2013eja can be interpreted as a probe of the electron EDM, with a small theoretical uncertainty Skripnikov ; Fleig:2014uaa . In contrast, nucleon, nuclear, and diamagnetic EDMs receive contributions from several operators, with varying levels of theoretical uncertainties. We provide the full expressions in Ref. SuppLE (including Refs. Gupta:2018lvp ; Alexandrou:2017qyt ; Pospelov:2000bw ; Lebedev:2004va ; Hisano:2012sc ; Demir:2002gg ; deVries:2012ab ; Pospelov:2001ys ; Bsaisou:2014zwa ; Dmitriev:2003sc ; deJesus:2005nb ; Ban:2010ea ; Dzuba:2009kn ; Sahoo:2016zvr ; Fleig:2018bsf ; Yamanaka:2017mef ; Yanase:2018qqq ; Dobaczewski:2018nim ; Benzke:2010tq ).

Matrix elements connecting quark EDMs to nucleon EDMs are relatively well known Gupta:2018lvp , but contributions from quark CEDMs and the Weinberg operator suffer from larger uncertainties. In addition to nucleon EDMs, nuclear and diamagnetic EDMs are generated by $C\!P$-odd nuclear forces that, for the operators under consideration, are dominated by $C\!P$-odd one-pion exchange between nucleons. The sizes of the associated low-energy constants have been calculated with QCD sum rules Pospelov:2001ys , with $\mathcal{O}(100\%)$ hadronic uncertainty. In addition, the nuclear many-body matrix elements that determine diamagnetic EDMs involve sizable nuclear uncertainties.

Current experimental limits are summarized in Table 1, which also shows the limits on systems that are not yet competitive, but could provide interesting constraints in the future. EDM experiments on 225Ra and 129Xe atoms have already provided limits PhysRevLett.86.22 ; Parker:2015yka ; Sachdeva:2019rkt and are quickly improving. Plans exist to measure the EDMs of charged nuclei such as the proton and deuteron in electromagnetic storage rings Eversmann:2015jnk . The EDM measurements of light nuclei can be more reliably interpreted in terms of BSM operators than is the case for $d_{\rm Hg}$ as the nuclear theory is under solid theoretical control deVries:2011an ; Bsaisou:2014zwa .

The operators $O_{\tilde{W}}$ and $O_{\varphi\tilde{W}B}$ contribute to the $C\!P\!$ asymmetry in $B\rightarrow X_{s}\gamma$ and to $C\!P$-odd triple-gauge couplings that were probed at LEP. Concerning the $B\to X_{s}\gamma$ asymmetry, we employ the expressions derived in Ref. Benzke:2010tq and take the required SM Wilson coefficients, as well as the hadronic parameters, from the same work. The triple-gauge vertices induced by $O_{\tilde{W}}$ and $O_{\varphi\tilde{W}B}$ are of the form $W^{+}W^{-}\gamma$ and $W^{+}W^{-}Z$, which were constrained using angular distributions in $e^{+}e^{-}\rightarrow W^{+}W^{-}$ Abbiendi:2000ei ; Abdallah:2008sf . In the notation of Ref. Gounaris:1996uw we have, $\tilde{\lambda}_{Z}=\tilde{\lambda}_{\gamma}=-2m_{W}^{2}C_{\tilde{W}}$ and $\tilde{\kappa}_{Z}=-t_{w}^{2}\tilde{\kappa}_{\gamma}=4t_{w}^{2}m_{W}^{2}C_{\varphi\tilde{W}B}$, $t_{w}=\tan\theta_{w}$, which leads to Tanabashi:2018oca

[TABLE]

As shown in Table 2, these constraints have already been improved by the study of the $W^{+}W^{-}$ cross section at the LHC Aaboud:2019nkz , and are likely to improve further in the context of EWPTs anticipated at the HL-LHC Cepeda:2019klc ; Azzi:2019yne ; deBlas:2019rxi .

Analysis.—To constrain the Higgs–gauge operators, we use EDM limits and the $C\!P\!$ asymmetry in $B\rightarrow X_{s}\gamma$ as listed in Table 1, as well as the LEP constraints on triple-gauge couplings given in Eq. (2). Nuclear and hadronic EDMs as well as the $C\!P\!$ asymmetry are affected by significant theoretical uncertainties. We follow Ref. Cirigliano:2016nyn and present limits in a variety of cases: (i) the “central” scenario, in which we neglect all hadronic and nuclear uncertainties, (ii) the “Rfit” strategy, in which all hadronic and nuclear matrix elements are varied within their allowed ranges to minimize the $\chi^{2}$ value, and (iii) the “Gaussian” strategy, in which the theoretical errors are treated in the same way as statistical errors are. This last strategy provides a realistic estimate of the impact of the theoretical errors when these are under control. We start by discussing the limits derived in the central case, which reflects the maximal constraining power of the low-energy measurements, assuming a single operator is present at the scale $\mu=\Lambda$. We subsequently consider the impact of the theoretical uncertainties in the Rfit scenario, as well as a scenario in which multiple Higgs–gauge operators appear at the scale $\Lambda$.

Turning on a single operator at the scale $\Lambda$, we see from Table 2 that the low-energy limits are very stringent. The bounds on the operators with EW gauge bosons are dominated by the electron EDM, which constrains $v^{2}C_{\varphi\tilde{W},\varphi\tilde{B},\varphi\tilde{W}B,\tilde{W}}$ to be $\mathcal{O}(10^{-6})$, corresponding to a BSM scale of $\sim 100$ TeV, assuming $C_{i}=1/\Lambda^{2}$, or $10$ TeV, including a loop factor, $C_{i}=1/(4\pi\Lambda)^{2}$. The constraints from the neutron and 199Hg EDMs are weaker, at the permille level for $v^{2}C_{\varphi\tilde{W}}$ and $v^{2}C_{\varphi\tilde{W}B}$ and at the percent level for $v^{2}C_{\varphi\tilde{B},\tilde{W}}$. The bounds on $C_{\varphi\tilde{G}}$ and $C_{\tilde{G}}$ are dominated by the mercury EDM in the central case. For both operators, the large uncertainties on the matrix element of the Weinberg operator imply that the constraints weaken by an order of magnitude and become dominated by the neutron EDM when moving from the central to the Rfit strategy. In contrast, the limits on the EW operators are very similar when using the Rfit strategy, as they are dominated by the ThO measurement. The fourth column in Table 2 shows the current collider limits for comparison.111Here we considered only limits arising from genuine dimension-6 contributions to $C\!P$-violating observables (more information on the CMS limits Sirunyan:2019twz ; Sirunyan:2019nbs is provided in Ref. SuppCMS ). Constraints on $v^{2}C_{\tilde{G}}$ stemming from dimension-8 contributions to jet cross sections were considered in Ref. Hirschi:2018etq , and estimated to be $\mathcal{O}(10^{-2})$. These high-energy probes are less sensitive by four to six orders of magnitude for most of the couplings, while they are competitive with the EDM constraints on $v^{2}C_{\varphi\tilde{G}}$ in the Rfit approach.

To see the effects of turning on multiple operators at the scale $\Lambda$, we investigate a scenario in which all Higgs–gauge couplings are present at $\mu=\Lambda$, while keeping $C_{\tilde{G},\tilde{W}}(\Lambda)=0$. This allows us to directly compare the low-energy limits to those of Ref. Bernlochner:2018opw . In this case there is one free direction left unconstrained by EDM measurements, even when neglecting theoretical uncertainties. For our choice of $\mu_{0}=1$ TeV, this combination of couplings is given by $\sim 0.17\,C_{\varphi\tilde{B}}+0.86\,C_{\varphi\tilde{W}}+0.48\,C_{\varphi\tilde{W}B}$. EDM measurements are not sufficient to constrain all four dimension-6 operators simultaneously and the $C\!P\!$ asymmetry in $B\rightarrow X_{s}\gamma$ and LEP observables are needed to close the free direction. When treating the theoretical uncertainties in the Rfit or Gaussian approach, the constraints from $d_{\rm Hg}$ and $d_{n}$ are degenerate, leading to another free direction. These free directions can be closed by reducing the errors on the theoretical predictions of matrix elements, or by considering improved constraints on the EDMs in Table 1 and bounds on the EDMs of additional systems, such as the proton or deuteron. Improvements on these three fronts are expected on the same timescale as the LHC Run III and the HL-LHC, for which the limits in Ref. Bernlochner:2018opw were derived.

We therefore consider improved determinations of the matrix elements that were set as targets for the future in Ref. Chien:2015xha . We assign 25% uncertainties to the nucleon EDM induced by the $u$- and $d$-quark CEDMs, and $50\%$ uncertainties on the nucleon EDM from $C_{\tilde{G}}$, the $C\!P$-odd pion–nucleon couplings, and the nuclear structure matrix elements. These uncertainty goals are by no means unrealistic considering recent lattice and nuclear-theory efforts Bhattacharya:2018qat ; Syritsyn:2019vvt ; Kim:2018rce ; Rizik:2018lrz , and in some cases have already been attained Dobaczewski:2018nim . On the experimental side, we assume $|d_{n}|<1.0\cdot 10^{-27}$ $e$ cm, which will be probed at the PSI and LANL neutron EDM experiments Schmidt-Wellenburg:2016nfv ; Ito:2017ywc , and $|d_{\rm Ra}|<10^{-27}$ $e$ cm, well within reach of the ANL radium EDM experiment Bishof:2016uqx . On a longer timescale, storage ring searches of the EDMs of light ions have the potential to compete with the neutron EDM Eversmann:2015jnk , and we assume $d_{p},\,d_{d}<1.0\cdot 10^{-27}$ $e$ cm. For the $C\!P\!$ asymmetry in $B\rightarrow X_{s}\gamma$, Belle II will be sensitive to sub-percent values, $|A_{B\rightarrow X_{s}\gamma}|<4\cdot 10^{-3}$ Kou:2018nap .

A comparison of the projected limits of Ref. Bernlochner:2018opw to the combination of future EDM and $B\to X_{s}\gamma$ limits in the $C_{\varphi\tilde{W}}$–$C_{\varphi\tilde{G}}$ and $C_{\varphi\tilde{W}}$–$C_{\varphi\tilde{B}}$ planes is shown in Fig. 2 and in Table 3. The non-zero central values for the low-energy curves are driven by the LEP bound (2) on $C_{\varphi\tilde{W}B}$, which deviates from zero by $\sim 2\sigma$. The gray, orange, and purple bands assume the proposed differential measurements in $pp\to h+2j$ have been performed on $36$, $300$, and $3000$ fb*-1* of integrated luminosity, respectively, while the red band shows the limits from low-energy experiments. The figure shows that the collider observables could in principle probe the $C_{\varphi\tilde{W}}$ and $C_{\varphi\tilde{B}}$ couplings at a comparable level as the low-energy limits with $36$ and $3000$ fb*-1* of data, respectively, but become relevant only when delicate cancellations between different couplings occur. The low-energy constraints on the gluonic operator $C_{\varphi\tilde{G}}$, are expected to be more stringent than the projected limits from the HL-LHC by roughly two orders of magnitude, see Table 3.

The strong constraints that EDM experiments put on the parameter space will manifest themselves in correlations between observables at the LHC. For example, the electron EDM bound establishes correlations between $C_{\varphi\tilde{W}B}$, $C_{\varphi\tilde{W}}$, and $C_{\varphi\tilde{B}}$, as can be seen from the lower panel in Fig. 2. An observation of large $C\!P\!$ violation in the Higgs–gauge sector, of the size of the right column in Table 3, would then require a non-zero value for $C_{\varphi\tilde{W}B}$. In such a scenario one would therefore expect large effects in diboson production, induced by $C_{\varphi\tilde{W}B}$, to be consistent with EDM experiments.

We can finally relax the assumption $C_{\tilde{W},\tilde{G}}(\Lambda)=0$, and consider all the $C\!P$-violating operators expected in the framework of universal theories. As argued above, the dominant EDM constraints are only sensitive to two linear combinations of the weak couplings $C_{\varphi\tilde{B}}$, $C_{\varphi\tilde{W}}$, $C_{\varphi\tilde{W}B}$, and $C_{\tilde{W}}$, so that EDM experiments could, in total, provide four independent constraints on the six operators in Eq. (1). One possible strategy to close the open directions in parameter space relies on the $C\!P\!$ asymmetry in $B\rightarrow X_{s}\gamma$ and/or LEP observables, but of course complementary LHC measurements would also provide the remaining two constraints. In either case, one again expects strong correlations between $C\!P$-violating observables in the Higgs and weak boson sectors, which illustrates the enormous potential of the low-energy probes in constraining the $C\!P$-odd sector of universal theories.

Conclusions.—In this Letter, we have analyzed the complementarity of LHC searches and low-energy experiments in constraining or discovering $C\!P\!$ violation in Higgs–gauge interactions, in the context of universal theories. In particular, we studied quantitatively the impact of EDMs on the allowed parameter space. Our work shows that despite the loop suppression EDMs cannot be neglected (as in recent LHC analyses)—in fact in a single-operator analysis there is very little room for observing $C\!P\!$ violation in the Higgs sector at the LHC. In a global analysis, flat or weakly bound directions from low-energy constraints are still possible, defining which additional operator combinations are most useful to be constrained by the (HL-)LHC, via the observables considered in Refs. Sirunyan:2019twz ; Sirunyan:2019nbs ; Aaboud:2017fye ; Bernlochner:2018opw and, potentially, EWPTs. Several lessons from our analysis extend beyond universal theories, where more $C\!P$-violating effective couplings appear. In this case EDMs enforce strong correlations among Higgs–gauge and other $C\!P$-violating couplings, which require either intricate cancellations and therefore insight on the new sources of $C\!P\!$ violation, or strong bounds on all the individual couplings.

Acknowledgments

Acknowledgements.

We thank Andrei Gritsan and Heshy Roskes for communication regarding Refs. Sirunyan:2019twz ; Sirunyan:2019nbs and Uli Haisch for discussions. This research is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contracts DE-AC52-06NA25396, DE-FG02-00ER41132, and DE-SC0009919. AC is supported by a Professorship Grant (PP00P2_176884) of the Swiss National Science Foundation. JdV is supported by the RHIC Physics Fellow Program of the RIKEN BNL Research Center.

Supplemental Material

Renormalization group evolution.—Here we provide additional details about the RG evolution and threshold corrections associated with the $C\!P$-violating Higgs–gauge interactions. We also give detailed expressions for the low-energy observables used to set constraints on the Wilson coefficients of the operators in Eq. (1) of the main text. These Higgs–gauge operators induce the following operators through the diagrams shown in Fig. 1

[TABLE]

with

[TABLE]

where $F^{\mu\nu}$ and $Z^{\mu\nu}$ are the photon and $Z$ field strengths, $t^{a}$ are the generators of $SU(3)_{c}$, and $Q_{f}$ and $T^{3}_{f}$ represent the electric charge and third component of weak isospin. The flavor-changing gluonic dipole $O_{8}$ is further suppressed by $\alpha_{\rm em}/(4\pi)$ with respect to $C_{7}$, and can be safely neglected. We use $e=-gs_{w}=-g^{\prime}c_{w}$ and $t_{w}=s_{w}/c_{w}$, with $s_{w}=\sin\theta_{w}$, $c_{w}=\cos\theta_{w}$, and the weak mixing angle $\theta_{w}$. The Wilson coefficients of the dipole operators in Eq. (3) are complex, we write $C_{\alpha}^{(f)}=c_{\alpha}^{(f)}+i\tilde{c}_{\alpha}^{(f)}$ with $c_{\alpha}^{(f)}$ and $\tilde{c}_{\alpha}^{(f)}$ real.

In addition to $B\rightarrow X_{s}\gamma$ as described by $\mathcal{L}_{b\to s}$, $C_{\varphi\tilde{W}B}$ and $C_{\tilde{W}}$ induce $b\rightarrow d\gamma$ and $s\rightarrow d\gamma$ dipoles via the same diagrams that contribute to $\mathcal{L}_{b\to s}$. The constraints from the direct $C\!P\!$ asymmetry in $B\rightarrow X_{d}\gamma$ and $B\rightarrow X_{s+d}\gamma$ are, respectively, weaker than and degenerate with those from $B\rightarrow X_{s}\gamma$. Similarly, constraints from $C\!P$-violating observables in kaon physics, such as $K_{L}\rightarrow\pi^{0}e^{+}e^{-}$ are not competitive, and for these reasons we concentrate on $B\rightarrow X_{s}\gamma$.222Further complementary constraints could arise from $b\to s\ell\ell$ and $B_{s}\to\mu\mu$ Bobeth:2015zqa . Altogether, we find the following matching conditions at $\mu_{t}=m_{t}\sim m_{H}$

[TABLE]

with $x_{t}=m_{t}^{2}/m_{W}^{2}$ and loop functions He:1993hx

[TABLE]

The anomalous dimensions that determine the RG evolution can be extracted from the $\log\mu_{t}$ terms in Eq. (5). We have checked that these agree with results in Refs. Dekens:2013zca ; Alonso:2013hga . The non-logarithmic terms in the matching relations depend on the regularization scheme. Our result is valid in both naive dimensional regularization with anticommuting $\gamma_{5}$ Chanowitz:1979zu and in the ’t Hooft–Veltman scheme tHooft:1972tcz . In both cases the Levi-Civita tensor was considered as an external 4-dimensional object. We use the RG evolution and the matching contributions to calculate the Wilson coefficients of the low-energy operators in Eq. (3) at the EW scale. We take into account that $\tilde{c}_{\gamma}^{(q)}$, $\tilde{c}_{g}^{(q)}$, and $C_{\tilde{G}}$ renormalize under QCD Weinberg:1989dx ; Wilczek:1976ry ; Braaten:1990gq ; Degrassi:2005zd . We then evolve the low-energy operators to the scale where QCD becomes non-perturbative, $\mu=\Lambda_{\chi}=2$ GeV. At the bottom, charm, and strange thresholds, the Weinberg operator obtains contributions analogous to the one in Eq. (5). The resulting fermion EDMs, CEDMs, the Weinberg operator, and $C_{7}$ at the scale $\mu=\Lambda_{\chi}$ are given in Table 4, where we assumed the initial scale $\mu_{0}=\Lambda=1$ TeV.

As explained in the main text, the weak operators $C_{\varphi\tilde{B}}$, $C_{\varphi\tilde{W}}$, $C_{\varphi\tilde{W}B}$, and $C_{\tilde{W}}$ contribute to the fermion EDM $\tilde{c}_{\gamma}^{(f)}$ almost exclusively via two combinations, proportional to $T^{3}_{f}$ and $Q_{f}$, respectively. Some sensitivity to the linear combinations of weak operators that do not contribute to $\tilde{c}_{\gamma}^{(f)}$ arises when one considers additional EW loops or matching into $C\!P$-violating semileptonic operators. The constraints are, however, weaker than the $B\rightarrow X_{s}\gamma$, LEP, and future collider constraints, so that we do not consider such effects here.

Low-energy observables.—In this section we provide explicit expressions for the low-energy observables used to constrain the Higgs–gauge operators. A more thorough discussion of all contributions and their uncertainties can be found in Ref. Dekens:2018bci . We begin with the constraint on the electron EDM. The most stringent constraint is set by the ACME collaboration Baron:2013eja ; Andreev:2018ayy using the polar molecule ThO. In principle, the electron spin-precession frequency receives contributions from both the electron EDM and $C\!P$-odd electron–nucleon interactions. The latter gets negligible contributions from the Higgs–gauge interactions under consideration, and we interpret the ThO measurement as a limit on the electron EDM using the relation Skripnikov ; Fleig:2014uaa

[TABLE]

and the experimental limit $\omega_{\text{ThO}}<1.3\,\mathrm{mrad}/\mathrm{s}$ at $90\%$ C.L. Baron:2013eja ; Andreev:2018ayy .

The neutron, $d_{n}$, and proton, $d_{p}$, EDMs are induced by quark (C)EDMs and the Weinberg operator. Contributions from first-generation EDMs are known with few percent accuracy Gupta:2018lvp . The contribution of the strange EDM has a larger uncertainty Gupta:2018lvp ; Alexandrou:2017qyt . QCD sum-rule calculations determine the contributions from the up- and down-quark CEDMs with roughly $50\%$ uncertainty, while the strange CEDM is assumed to vanish in the PQ scenario Pospelov:2000bw ; Lebedev:2004va ; Pospelov:2005pr ; Hisano:2012sc . Contributions from the Weinberg operator are difficult to determine and current estimates from QCD sum-rules Demir:2002gg and naive dimensional analysis Weinberg:1989dx have an $\mathcal{O}(100\%)$ uncertainty

[TABLE]

where $d_{q}=Q_{q}m_{q}\tilde{c}_{\gamma}^{(q)}$ and $\tilde{d}_{q}=m_{q}\tilde{c}_{g}^{(q)}$.

EDMs of diamagnetic atoms and light nuclei receive contributions not only from the nucleon EDMs, but also from the $C\!P$-violating nucleon–nucleon potential. For the operators under consideration, this potential is dominated deVries:2012ab by one-pion-exchange contributions involving the $C\!P$-odd pion–nucleon ($\pi N$) vertices

[TABLE]

in terms of the Pauli matrices $\tau$, the nucleon doublet $N=(p\,\,n)^{T}$, and the pion triplet $\pi$. The sizes of $\bar{g}_{0,1}$ have been calculated with QCD sum rules Pospelov:2001ys

[TABLE]

In combination with $C\!P$-even nuclear forces and currents, the nucleon EDMs and the $C\!P$-odd $\pi N$ interactions can be used to calculate the EDMs of light nuclei. In particular, the EDM of the deuteron is given by Bsaisou:2014zwa

[TABLE]

The EDM of the diamagnetic atom 199Hg gets contributions from both nuclear and leptonic $C\!P$-odd interactions. For our purposes we use Dmitriev:2003sc ; deJesus:2005nb ; Ban:2010ea ; Dzuba:2009kn ; Engel:2013lsa ; Sahoo:2016zvr ; Fleig:2018bsf

[TABLE]

neglecting semileptonic interactions Yamanaka:2017mef ; Engel:2013lsa ; Yanase:2018qqq that receive small contributions from the Higgs–gauge operators. Due to octopole deformations, the EDM of 225Ra is dominated by the pion-exchange contributions. We use Dobaczewski:2018nim

[TABLE]

Although it has little affect in our analysis, for completeness we provide the following expression for 129Xe Dzuba:2009kn ; Engel:2013lsa

[TABLE]

Finally, we employ the expressions of Ref. Benzke:2010tq for the $C\!P\!$ asymmetry in $B\to X_{s}\gamma$

[TABLE]

where $\epsilon_{s}=\frac{V_{us}^{*}V_{ub}}{V_{ts}^{*}V_{tb}}$ and we use

[TABLE]

The Wilson coefficients are given by

[TABLE]

CMS limits.—Refs. Sirunyan:2019twz ; Sirunyan:2019nbs consider limits on $C\!P$-violating effective couplings that affect the production of a Higgs boson via vector-boson fusion (VBF), with the Higgs subsequently decaying into $\tau\tau$ or four leptons. The observables discussed in Ref. Sirunyan:2019twz ; Sirunyan:2019nbs are mostly sensitive to the modification of the $hZZ$ vertex, which affects both VBF and $h\rightarrow 4l$, and of the $hWW$ vertex, which affects VBF. The bounds in Sirunyan:2019twz ; Sirunyan:2019nbs are expressed in terms of the anomalous coupling $a_{3}$, parameterizing $C\!P$-violating contributions to the $h\rightarrow ZZ$ vertex, and $r_{a3}$, the ratio of the $CP$-violating $hWW$ and $hZZ$ vertices. The coefficients of the SMEFT operators defined in Eq. (1) of the manuscript can be mapped onto the effective couplings $a_{3}$ and $r_{a3}$ as follows

[TABLE]

where $a_{1}$ denotes the SM $hZZ$ coupling. In the analyses of Refs. Sirunyan:2019twz ; Sirunyan:2019nbs , $r_{a3}$ is, for convenience, set to one. The observed 95% C.L. constraints in Ref. Sirunyan:2019nbs are however insensitive to the value of $r_{a3}$  CMSprivate , and can thus be interpreted as a bound on $a_{3}/a_{1}$ as given in Eq. (18).

Ref. Sirunyan:2019nbs presents a strong $68\%$ C.L. limit, which is dominated by corrections to VBF, and almost reaches the 2$\sigma$ level. In this case, the results in Ref. Sirunyan:2019nbs depend on the choice of $r_{a3}$, but, being dominated by VBF, the constraints obtained with $r_{a3}=1$ can be converted into constraints with arbitrary $r_{a3}$ Sirunyan:2019twz ; CMSprivate . As an estimate of the bounds that can be reached in the near future, we quote the expected 95% C.L. limit in Ref. Sirunyan:2019nbs , which gives

[TABLE]