Higgs probes of axion-like particles

We study axion-like particle contributions to the Higgs boson decays. The particle is assumed to couple with the standard model electroweak gauge bosons. Although direct productions of axion-like particles have often been discussed, we investigate indirect contributions to the Higgs boson decays into fermions, photons, W, and Z bosons at the one-loop level. It is found that the corrections to the fermions are suppressed, whereas precise measurements of the di-photon channel of the Higgs boson decay can provide a significant probe of the model especially when the axion-like particle is heavy and its coupling to di-photon is suppressed.


Introduction
The introduction of an axion-like particle (ALP) is one of the simplest extensions of the Standard Model (SM) and has been attracting wide interest.It is often motivated in models with spontaneous violations of global symmetries and is characterized by interactions with gauge bosons.
In this paper, we study ALP contributions to decay widths of the Higgs boson (h).In literature, direct ALP productions from Higgs boson decays, e.g., h → Za and h → aa, have often been discussed [4,[21][22][23][24].Those channels are open when the ALP is lighter than the Higgs boson.However, if it is heavier, the direct production channels are forbidden kinematically, and the ALP affects the Higgs boson decays via radiative corrections.In this paper, we will investigate ALP-loop contributions to the Higgs boson decays: h → f f with f denoting SM fermions, h → γγ, h → Zγ, h → ZZ * , and h → W W * .They will be compared with the direct ALP productions from Higgs boson decays.We will also take collider constraints into account.It will be shown that the contribution to the fermion channels is suppressed, while h → γγ provides a significant probe of the ALP compared with other channels such as h → Zγ, h → ZZ * , h → W W * , and h → aa.
This paper is organized as follows.In Sec. 2, we introduce the ALP model.In Sec. 3, the ALP contributions to Higgs boson decays are explained.Numerical results are given in Sec. 4, and Sec. 5 is devoted to the conclusion.In Appendix A, the analytic expressions for the ALP contributions at the one-loop level are summarized.

Model
Let us introduce ALP (a).It is assumed to couple with the SU(2) L gauge boson (W a µ ) and the U(1) Y gauge boson (B µ ).We do not consider ALP couplings with other SM particles.The Lagrangian is given by where m a and f a are the mass and decay constant of the ALP, respectively.In this paper, the coupling constants, c W W and c BB , as well as m a and f a are regarded as free parameters but m a < f a is satisfied.Field strengths of the gauge bosons and their dual are shown as where g is the SU(2) L gauge coupling constant.The totally antisymmetric tensors are defined with ϵ 012 = 1 and ϵ 0123 = 1.After the EW symmetry breaking, the above interactions are expressed as where quartic interaction terms are omitted, and the field strengths denote Here and hereafter, A µ , Z µ , and W µ represent the photon, Z, and charged W bosons, respectively.Those coupling constants are related to c W W and c BB as ) ) (2.9) Here and hereafter, c W = cos θ W , s W = sin θ W , c 2W = cos 2θ W , and s 2W = sin 2θ W with the Weinberg angle θ W .It is stressed that the original ALP interactions in our setup are governed by two coupling constants, c BB and c W W , though there are four couplings, g aγγ , g aZγ , g aZZ , and g aW W , after the EW symmetry breaking.It is often useful to represent g aZγ and g aZZ in terms of g aγγ and g aW W , ) (2.11)

Higgs boson decay
In this section, we explain ALP contributions to the Higgs boson decays at the oneloop level.The decay channels in interest are h → f f , h → ZZ * , h → W W * , h → γγ, and h → Zγ as well as h → aa.#1 Within the SM, the first three of them proceed at the tree level, and the following two are induced radiatively.Also, the last channel is absent.The ALP affects all of them via radiative corrections.
For the radiative corrections, we adopt the on-shell renormalization to the EW sector by following Refs.[25,26] #2 .The counterterms relevant to ALP corrections to the Higgs boson decays are introduced in Sec.3.1, and the decay width for each channel will be given in Sec.3.2, 3.3, and 3.4.
#1 It is noted that there is no ALP contribution to h → gg at the one-loop level in our setup.Also, h → Za is not induced at this level because gauge-boson loop contributions vanish [4].#2 Since there is no ALP mixing with the CP-odd Nambu-Goldstone boson at the one-loop level, we use the gauge fixing terms, which are the same as in the SM.

Renormalization
The bare gauge fields (described with the subscript B) are related to the renormalized ones by means of the counterterms (denoted as δX) as The renormalized W, Z masses and the measured value of the fine-structure constant are related to bare parameters as The Weinberg angle and the Higgs vacuum expectation value (VEV) satisfy the following tree-level relations, Then, their counterterms are obtained as ) Let us represent the renormalized inverse propagators of gauge bosons as with V, V ′ = W, Z, γ.The self-energies Π µν V V ′ (p 2 ) are decomposed into longitudinal (L) and transverse (T ) polarization components, Since the longitudinal component is irrelevant to the following discussion, we focus on the transverse one and omit the subscript, T and L, for simplicity.#3 In terms of the 1PI contributions and counterterms, they are given by #4 #3 Since the ALP is not coupled with the Higgs boson and fermions at the tree level, there are no ALP contributions to the Higgs and fermion self-energies.#4 Since there is no ALP contribution to tadpoles, we omit the tadpole terms in this paper. (3.12) In the renormalization scheme explored by Refs.[25,26], the residue of the two-point function for the weak gauge bosons is not unity.Hence, we need to add a derivative of the self-energy, Π V V (m 2 V ) ′ , to transition amplitudes with external W ± and Z bosons, Let us focus on the ALP contributions.They arise via Π 1PI V V ′ , and the results are obtained in Appendix A.2.In the following, the counterterms denote those from ALP, δX = δX ALP .In addition, we use the relation, Π 1PI V V ′ (0) ALP = 0 (see Appendix A.2) to simplify the expressions.Then, we obtain ) yielding δs 2 W and δv (see Eqs. ) (3.17) In our calculation, the Higgs VEV v is represented by the measured value of the Fermi coupling constant G F , where ∆r is obtained from the muon decay rate evaluated at the one-loop level [27].
In our setup, it satisfies the following relation, The decay proceeds at the tree level in the SM, and the ALP affects it via radiative corrections.In general, the renormalized hf f vertex is decomposed as where p 1 (p 2 ) denotes the incoming four-momentum of the fermion (anti-fermion), and q is the outgoing four-momentum of the Higgs boson (see the left panel of Fig. 1).The form factors are composed of the tree and one-loop contributions as The former is given by The latter is composed of the 1PI and counterterm contributions, In our setup, there is no ALP contribution to the 1PI terms at the one-loop level, and the counterterms do not vanish only for δΓ S hf f in the presence of the Higgs VEV counterterm.Thus, the ALP contributions are obtained as As a result, the decay rate for At the leading order, the rate is obtained at the tree level as with N f c = 3 (1) for f being quarks (leptons).The kinetic factor is defined as The SM correction ∆ f SM is found in Refs.[28,29], and the ALP term ∆ f ALP at the one-loop level is obtained from Eqs. (3.19) and (3.24) as Therefore, ∆ f ALP is canceled out, and there is no correction from ALP to h → f f at this level.
The decay proceeds at the tree level in the SM, and the ALP affects it via radiative corrections.In general, the renormalized hV V vertex is decomposed as where p µ 1 and p µ 2 are the incoming four-momenta of the weak bosons, and q µ is the outgoing four-momentum of the Higgs boson (see the right panel of Fig. 1).The form factors are composed of the tree and one-loop contributions as The former is given by The latter is decomposed into the 1PI and counterterm contributions as #5 The ALP contributes to Γ 1,1PI hV V and Γ 2,1PI hV V at the one-loop level.The results are given in Appendix A.3.The ALP contributions to the counterterms become non-zero only for δΓ 1  hV V in the presence of the Higgs VEV counterterm as Since the Higgs boson mass is smaller than 2m Z , one of the final-state Z bosons is off-shell and decays into a pair of SM fermions.The decay rate for h → ZZ * → Zf f is written as # 5 The ALP contribution to the tadpole is absent and omitted in the expression.
In the following, we neglect masses of the fermions.At the leading order, the rate is obtained at the tree level as where , and a f = I f 3 /2 with the weak isospin I f 3 and the electric charge Q f .The radiative corrections ∆ Z SM/ALP can be written in terms of the renormalized quantities [30,31].The ALP correction ∆ Z ALP is obtained as #6 The kinematic factor λ(x, y) is defined by λ(x, y) = 1 − x − y 2 λ(x, y) λ(x, y) + 12xy . (3.38)

h → W W *
Similarly to h → ZZ * , one of the final-state W bosons is off-shell and decays into a pair of SM fermions.The decay rate for h In the following, we neglect masses of the fermions.At the leading order, the rate is obtained at the tree level as # 6 In general, there are also corrections from renormalized Zf f and hf f vertices and 1PI box diagrams (see Refs. [30,31]).However, they do not receive ALP contributions in our setup and are omitted in the expression.

< l a t e x i t s h a 1 _ b a s e 6 4 = "
w l b c C P 2 y t G 7 F y P R K q x E 3 9 5 5 s r d y c 3 k m v E S e k x / o / x n Z J e 9 x B W 7 7 p / V i i S 0 / P c S P h y 4 6 m I l 2 L z h k 7 8 J 4 h 5 u w A 7 P o h q H G R S X F b x 2 i 4 9 U P H u Z w s G Y U 9 f n i t a W r 0 w u 3 0 4 M e h w t w E X J 4 m t d h A e 7 C I q y C p b The squared tree-level amplitude |M W 0 | 2 is obtained as The radiative corrections ∆ W SM/ALP can be written in terms of the renormalized quantities [31].The ALP correction ∆ W ALP is obtained as #7 Note that since there is no photon-mediated diagram, we have no infrared (IR) divergences in ALP contributions.Thus, the treatment of IR regularization in the numerical analysis is the same as that in the SM [31].
In contrast to the previous channels, h → V γ proceeds via radiative corrections in the SM as well as the ALP model.The renormalized hVV ′ vertex is expressed as with V = γ, Z and V ′ = γ.Here, p µ 1 and p µ 2 are the incoming four-momenta of the gauge bosons, and q µ is the outgoing four-momentum of the Higgs boson (see the left panel of Fig. 2).The form factors are composed only of the one-loop contributions, which are generally decomposed into the 1PI and counterterm components as Similarly to h → ZZ * , there are generally corrections from renormalized W f f ′ and hf f vertices and 1PI box diagrams.However, ALP contributions vanish in our setup, and only non-zero terms are shown.
Since the Z-γ mixing cannot induce h → γγ at the one-loop level, the counterterms for the hγγ vertex are zero.However, those for hZγ generally exist as In the present setup, it is found from Eq. (3.16) that the ALP contribution to δΓ 1 hZγ is canceled out.Therefore, the ALP contributions are obtained by the 1PI terms as The results are obtained in Appendix A.3.

h → γγ
The decay rate for h → γγ is given by with τ i = m 2 h /(4m 2 i ) for i = W, f, a.The SM loop functions are given by [32][33][34] with f (τ ) defined as The ALP loop function is given by The term ∆ f QCD denotes QCD corrections to the quark-loop contributions.We include it up to next-to-next-to-leading order, and whose expressions in the large topmass limit (τ t ≪ 1) are given in Refs.[35,36].

h → Zγ
The decay rate for h → Zγ is given by with The SM loop functions are given by [34,37,38] with I 1 (τ, λ) and I 2 (τ, λ) defined as The function f (τ ) is found in Eq. (3.50), while g(τ ) is defined as The ALP loop function is given by The term ∆ f QCD denotes QCD corrections to the quark-loop contributions.In our analysis, it is included up to the next-to-leading order, and its expression is the same as that for the di-photon decay [39].The ALP mass is chosen to be m a = 5, 100, 500, 800 GeV (solid blue, yellow, green, and red lines, respectively).The horizontal dotted/dashed lines are experimental bounds and future sensitivities.
h → γγ becomes twice as large as or even larger than the SM contribution for g aγγ ∼ 10 TeV −1 .
In the plots, experimental bounds and future prospects are shown by horizontal lines.The dashed lines denote the bounds obtained by analyzing the full dataset of the ATLAS Run II [42].The CMS experiment provides almost the same results [43].As for the prospects, the HL-LHC sensitivities [44] are shown by the dot-dashed lines.We also show the dotted lines that represent far-future sensitivities expected by combining data from the FCC-ee/eh/hh experiments [45].
By comparing the ALP contributions with the experimental bound/sensitivities, it is found that h → γγ provides the best probe to the ALP among the decay channels.The model is excluded for g aγγ ≳ 2.8 − 3.9 TeV −1 , depending on the ALP mass, except for the narrow region at g aγγ ∼ 10 TeV −1 .The HL-LHC experiment can probe the ALP with g aγγ ≳ 1.6 − 2.2 TeV −1 , and the sensitivity may reach g aγγ = 0.7 − 0.9 TeV −1 at FCC-ee/eh/hh.In Fig. 4, we study the case with g aW W ̸ = 0 instead of g aγγ .Here, κ W , κ Z , κ γ , κ Zγ are shown as a function of g aW W with setting g aγγ = 0.It is found that the ALP contributions are less sensitive to m a and become effective for g aW W ≳ 1 TeV −1 in h → γγ, Zγ and 10 TeV −1 in h → ZZ * , W W * .The ALP contributions decrease the decay width of h → W W * and increase that of h → ZZ * .The difference between these two behaviors is understood by noticing that the hZγ vertex and the Z-γ mixing give positive corrections in h → ZZ * .On the other hand, the ALP contributions decrease the decay widths of h → γγ and h → Zγ for g aW W ≲ 10 TeV −1 .However, the magnitude can become comparable to or larger than the SM corrections when g aW W ≳ 10 TeV −1 and increase the decay widths compared with the SM predictions.
By comparing the ALP contributions with the experimental bound/sensitivities, it is found that h → γγ provides the best probe to the ALP, similarly to the case with g aγγ ̸ = 0. Since g aZγ ∝ (g aW W − g aγγ ) is satisfied (see Eq.(2.10)), the values of κ γ in Fig. 4 become exactly the same as those in Fig. 3. Therefore, we obtain the same conclusion as the case with g aγγ ̸ = 0.
Let us next compare the Higgs results with collider constraints.In Fig. 5, we show the collider limits on the (g aγγ , g aW W ) plane.Here, we consider m a = 5, 195, and 600 GeV, where the constraints on g aγγ is relatively weak (see the discussion in Ref. [20]).The regions excluded by the collider limits are filled with gray color surrounded by dashed lines.For m a ≲ 100 GeV, the LEP experiments give strong bounds, while heavier ALPs are constrained by the LHC results.The details of the collider constraints are given in Ref. [20].It is commented that h → aa may have a sensitivity to the parameter space for m a = 5 GeV.However, we found that the effect is so weak that no lines appear in the parameter range of Fig. 5.
In the plots, the experimental bound and future sensitivities to the ALP contributions to h → γγ are shown by solid red (ATLAS Run II bound), green (HL-LHC), and blue (FCC-ee/eh/hh) lines.For m a = 5 GeV, it is found that the h → γγ sen- sitivities are almost weaker than the LEP constraints even at FCC-ee/eh/hh.By contrast, for m a = 195 GeV the future sensitivity at HL-LHC can compete with the LHC constraint from the measurement of pp → ννγγ, and the model could be probed better by FCC-ee/eh/hh.For m a = 600 GeV, it is noticed that the collider constraints are relaxed especially for g aγγ ∼ 0, and thus, h → γγ provides a significant probe of the model.
Since the model is much less constrained for g aγγ ∼ 0, let us discuss the ALP mass dependence of the results for g aγγ = 0 and compare them with the collider constraints.We also take flavor constraints into account for m a ≲ 4.8 GeV (see Ref. [20] for details).In Fig. 6, we show the experimental bound and future sensitivities on g aW W by the measurements of κ γ (solid lines) as a function of m a .It is found that the current bound on κ γ (red line) provides a stronger constraint than the colliders; the region of g aW W ≳ 3.2 − 4.1 TeV −1 is excluded for m a > 300 GeV.It is noted that there is a narrow allowed region above g aW W = 10 TeV −1 between the solid lines, where the ALP contribution is almost twice as large as the SM one with an opposite sign.As for future prospects, g aW W > 1.7 − 2.3 TeV −1 can be probed for m a > 180 GeV by HL-LHC (green line).If FCC-ee/eh/hh would be constructed, the sensitivity could reach g aW W = 0.7 − 1 TeV −1 (blue line) and might compete with or even be better than the collider constraints at LEP and LHC.
In the figure, we also show the result of h → aa, which is open kinematically for m a < m h /2.The current bound and future sensitivities are shown by the dot-dashed red, green, and blue lines, respectively.It is found that the decay is less sensitive to the ALP model compared with h → γγ.
Before closing this section, let us comment on the case for g aW W = 0.In Fig. 6, g aW W was varied with fixing g aγγ = 0.If g aγγ is varied with g aW W = 0, we obtain the same lines for h → γγ as those in Fig. 6 (see Eq.(2.10)).However, the collider constraints are much more severe, and almost the whole parameter regions accessible by future h → γγ measurements are already excluded.Similarly, if we consider the case for g aZγ = 0, i.e., g aγγ = g aW W is fixed, the ALP contributions to h → γγ and h → Zγ are suppressed.Then, the ALP is probed by h → ZZ * and h → W W * .Since the ALP contributions become effective only for g aγγ ∼ 10 TeV −1 , such a parameter region is excluded by the collider constraints.

Conclusion
In this paper, we have studied the Higgs boson decays in the ALP model.The ALP is assumed to couple with the SM SU(2) L and U(1) Y gauge bosons.We have provided the formulae for the ALP contributions to h → f f , h → γγ, h → Zγ, h → ZZ, and h → W + W − as well as h → aa at the one-loop level.It was found that those to h → f f vanish at this level, while the other decay widths receive ALP contributions via radiative corrections.
Among the decay channels, it was concluded that h → γγ can provide a significant probe of the model.Compared with the collider constraints, the decay becomes relevant especially when the ALP is heavy and g aγγ is suppressed.For g aγγ = 0, it was shown that the current LHC result of h → γγ excludes the ALP with g aW W ≳ 3.2 − 4.1 TeV −1 and m a > 300 GeV.In the future, the HL-LHC experiment can probe g aW W > 1.7 − 2.3 TeV −1 for m a > 180 GeV.Also, the sensitivity could reach g aW W = 0.7 − 1 TeV −1 at the FCC-ee/eh/hh experiments; it might compete with or even be better than the collider constraints on the direct ALP productions at LEP and LHC.Moreover, h → γγ has a better sensitivity than h → aa.Therefore, the precision measurements of h → γγ could provide a powerful tool in searching for the ALP model.

Figure 1 :
Figure 1: Momentum assignments to the Higgs boson decays.

r 1 4 8 N
A P a Z s A I b P L A B N 2 d 9 b d g Q g b b p 6 8 e v C g H j T x Y P w Z X v w D m v Q n G I 8 1 8 e L B d z 9 I Q Q r O Z m f e e d 7 n e e e Z e X X H 5 J 4 k 5 G h J O Z M 4 u 3 x u 5 X z y w s

p 5 j 4 u 6 Figure 2 :
Figure 2: Feynman diagram of ALP contributions to h → γγ, Zγ and h → aa.In the right panel, the gauge bosons denote W/Z/γ.

Figure 4 :
Figure 4: Same as Fig 3 but g aW W is varied with g aγγ = 0.

Figure 5 :
Figure 5: Experimental bound (red) and future sensitivities (green and blue) of ALP contributions to h → γγ (solid lines) on the (g aγγ , g aW W ) plane.The gray regions surrounded by dashed lines are excluded by the collider constraints.

Figure 6 :
Figure 6: Experimental bound (red) and future sensitivities (green and blue) of ALP contributions to h → γγ (solid lines) and h → aa (dot-dashed lines) as a function of the ALP mass.The gray regions surrounded by dashed lines are excluded by the collider constraints and/or flavor limits.