Diphoton Excess and Running Couplings

The recently observed diphoton excess at the LHC may suggest the existence of a singlet (pseudo-) scalar particle with a mass of 750 GeV which couples to gluons and photons. Assuming that the couplings to gluons and photons originate from loops of fermions and/or scalars charged under the Standard Model gauge groups, we show that here is a model-independent upper bound on the cross section $\sigma(pp\to S\to \gamma\gamma)$ as a function of the cutoff scale $\Lambda$ and masses of the fermions and scalars in the loop. Such a bound comes from the fact that the contribution of each particle to the diphoton event amplitude is proportional to its contribution to the one-loop $\beta$ functions of the gauge couplings. We also investigate the perturbativity of running Yukawa couplings in models with fermion loops, and show the upper bounds on $\sigma(pp\to S\to \gamma\gamma)$ for explicit models.


Introduction
Recently, the ATLAS and CMS collaborations reported an excess of diphoton events implying a resonance with a mass of around 750 GeV [1,2]. The ATLAS collaboration has 3.2 fb −1 of data, and the largest excess is found at around the diphoton invariant mass of m γγ 750 GeV with the local (global) significance of 3.6σ (2.0σ) for a narrow width case. When a large width for the signal component is assumed, the local (global) significance increases to 3.9σ (2.3σ) at the width of about 45 GeV. The CMS collaboration, with 2.6 fb −1 of data, also reported an excess at around m γγ 750 GeV with the local (global) significance of 2.6σ (1.2σ) for a narrow width case, while the significance does not increase with a larger width. Possible explanations and implications of this excess have been extensively discussed [3,4,5,6,7].
One of the plausible explanations of the excess is that a scalar or pseudoscalar particle S with a mass of 750 GeV is produced through gluon fusion and decays into a pair of photons, gg → S → γγ, via diagrams with new fermions and/or bosons charged under the Standard Model (SM) gauge groups running in the loops [3,4,5,6]. In order to explain the excess with perturbative couplings, however, the new particles in the loop should have large quantum numbers and/or large multiplicity, which implies that the perturbativity of the SM gauge groups may break down at some high scale below the Planck scale. In this letter, we address this issue and investigate the perturbativity of such models.
Our main conclusions are as follows: 1. We point out that the contribution of each particle in the loop to the diphoton event amplitude is proportional to its contribution to the one-loop β functions of the gauge couplings at the leading order, independently of the representations of the particles in the loop. Consequently, there is a generic upper bound on the cross section σ(pp → S → γγ) as a function of the cutoff scale Λ and masses of the fermions and scalars in the loop. We also numerically evaluate such a bound, taking into account the following constraints: (i) the constraints from Landau pole, requiring that the gauge couplings remain perturbative up to the scale Λ, and (ii) the constraint from the scale dependence of the strong coupling constant based on the LHC [8].
2. We also investigate the running of the Yukawa coupling in models with fermion loops. The upper bound on σ(pp → S → γγ) is presented as a function of the fermion mass and the cutoff scale Λ for some explicit models with vector-like quarks.
The generic analysis in the first part, which can be applied to models with fermions and scalars in the loop in arbitrary representations, was not considered in the previous works. The analysis of the second part is close to those of Ref. [3], where the authors investigated the running of the gauge, Yukawa, and scalar quartic couplings in models with multiple generations of fermions in the loop. (See also Refs. [4] for related works.) They considered several model points with fixed fermion masses and the number of generations. Our analysis is complementary in the sense that the fermion mass, the Yukawa coupling, the number of generations, as well as the cutoff scale are taken as free parameters.
In the next section, we investigate the running of gauge couplings in generic setup with fermions and scalars in arbitrary representations, and show that there is a model-independent upper bound on the cross section σ(pp → S → γγ). In Sec. 3, we investigate the running of the Yukawa coupling (as well as those of gauge couplings) in explicit models and present the upper bound on σ(pp → S → γγ) as a function of the fermion mass and the cutoff scale Λ. We also briefly discuss the LHC constraints on vector-like quarks, and comment on the running of the scalar quartic coupling of S. We conclude in Sec. 4.

Running gauge couplings and generic upper bound on the diphoton event rate
The reported diphoton excess can be explained by a new scalar particle S, with a mass of m S 750 GeV, which is produced by a gluon fusion and decays into two photons. The cross section is given by where √ s = 13 TeV is the center-of-mass energy of the LHC, and C gg = (π 2 /8) being the gluon parton distribution function. In our numerical calculation, we use the MSTW2008 NLO set [9] evaluated at the scale µ = m S , which gives C gg 2.1 × 10 3 . The reported excess [1,2] suggests σ(pp → S → γγ) ∼ O(1)-10 fb.
We assume that the production and the decay of the singlet scalar S is induced through loops of new fermions ψ i and/or scalars φ i . In order to make the analysis model-independent, we consider that they have generic quantum numbers (R (3) i , R (2) i , Y i ) under the SM gauge groups SU(3)×SU(2)×U(1) Y . The relevant part of the Lagrangian is given by 1 where η i = 1/2 for Majorana fermions and real scalars, and η i = 1 otherwise. (Notice that Majorana fermions and real scalars are possible only for the case of real representation of the SM gauge group, such as (8,1,0) and (1,3,0).) In the following, we assume CP-conservation, and consider the two cases of scalar S (y 5i = 0) and pseudoscalar S (y i = A i = 0) separately. The partial decay rates of S into gg and γγ are given by where in the case of scalar S, and in the case of pseudoscalar S. Here, d being singlet, fundamental representation, and adjoint representation, respectively. The trace of the electric charge squared is given by and the loop functions are (for τ < 1) with τ i = m 2 S /4m 2 i . (These loop functions are normalized so that they become 1 for τ → 0.) Let us now discuss the running gauge couplings of the SM for a scale at µ > m i , which are give by, at the one loop, where α a,SM (m i ) is evaluated by using the renormalization group (RG) equations of the SM, b SM a = (41/6, −19/6, −7), and Note that the contributions of each fermion or scalar to ∆b a in (13) are the same as the coefficients in the diphoton production rate, Eqs. (4)- (7). Therefore, by defining effective masses m eff i and its minimal value as one can obtain upper bounds on Γ(S → gg) and Γ(S → γγ) as functions of ∆b a and m eff min as follows; We consider the following two constraints on ∆b a : (i) Landau pole: We require that the SM gauge couplings are perturbative up to a scale Λ, 2 which leads to upper bounds on ∆b a as functions of Λ (and m i ).
(ii) Running α 3 : In addition, too large ∆b 3 (with relatively small m i ) modifies the evolution of the strong coupling constant and conflicts with the scale dependence of α 3 observed by the LHC [8]. We require that ∆b 3 is below the 2σ upper bound given in Ref. [8]. 3 For instance, the bound is ∆b 3 < 5.2 (15.9) when the mass of the particle in the loop is 500 (700) GeV.  14). Left: broad width case, Γ S,total = 45 GeV. Right: narrow width case, Γ S,total = Γ(S → gg) + Γ(S → γγ). The solid, dashed, and dotted lines show the contours of σ(pp → S → γγ) max = 10, 5, and 3 fb, respectively. The blue lines represent the case that S is a scalar, while red lines are for the pseudoscalar case.
These bounds on ∆b a lead to the maximal values of Γ(S → gg) and Γ(S → γγ) according to Eqs. (15) and (16), which are then converted to the upper bound on the cross section for the process pp → S → γγ. In particular, as one can see from Eq. (1), the cross section becomes larger as Γ(S → gg) increases. In addition, when Γ(S → gg) takes its largest possible value, the partial decay rates into electroweak gauge boson pairs are always much smaller than Γ(S → gg), and σ(pp → S → γγ) increases as Γ(S → γγ) becomes larger. Thus, with Λ and m eff min being fixed, the cross section takes its largest value when ∆b 1 , ∆b 2 and ∆b 3 are all maximized. Fig. 1 shows the upper bound on σ(pp → S → γγ) as a function of Λ and m eff min , which is obtained from Eqs. (1), (15), and (16). The left figure shows the case of fixed broad width Γ S,total = 45 GeV, while the right figure represents the case of narrow width, Γ S,total = Γ(S → gg) + Γ(S → γγ). 4 The red and blue lines show the cases that S is a scalar and a pseudoscalar, respectively. Here, for simplicity, we have taken m i = m eff min to calculate the running coupling with Eq. (12), and also to obtain the upper bound on ∆b 3 from Ref. [8]. 5 As can be seen in the left panel of Fig. 1, the cutoff scale Λ cannot be very large for a broad width case. Below the kink at m 600-700 GeV, the constraint from the α 3 (µ) measurement gives a severe upper bound on ∆b 3 . In this region, the upper bound on Λ is determined by the condition of Landau poles of α 1,2 . Above the kink, the Landau pole condition on ∆b 3 is stronger than that from the α 3 (µ) measurement. In the narrow width case shown in the right panel of Fig. 1, the bounds become weaker than the broad width case, but they still constrain the region of m eff min O(TeV) when the cutoff scale Λ is large. For instance, in order to have σ(pp → S → γγ) = 10 fb with Λ = 10 18 GeV (10 15 GeV), the effective mass should be m eff min 870 (1100) GeV in the case that S is a scalar, and m eff min 1300 (1700) GeV in the case of pseudoscalar. Before closing this section, several comments are in order.
• The bounds in Fig. 1 are very conservative, and they can become severer in concrete and realistic models. First of all, ∆b 1 , ∆b 2 and ∆b 3 are simultaneously maximized in Fig. 1, but it is not generically the case in concrete models. Secondly, the region with small mass and large ∆b a is severely constrained by the direct search for the new particles ψ i and φ i . For instance, in the broad width case, the constraint from the α 3 (µ) measurement for m = 600 GeV is about ∆b 3 < 8.7, and the upper bound corresponds to 13 Dirac pairs of vector-like quarks if they are in fundamental representations. Such a model is likely to be already excluded by direct searches, unless the new colored particles decay in a very complicated manner to escape from LHC searches. The direct search can constrain the model for the narrow width case as well. (See also the discussion in the next section.) Although it is difficult to saturate the bounds in Fig. 1 in concrete realistic models, they are model-independent and conservative, and yet constraining interesting regions of m eff min and Λ. Therefore the bounds in Fig. 1 can be an important first step to explore the physics behind the diphoton signal.
• In models with fermion loops, the Yukawa coupling y (5)i at low energy becomes typically smaller than unity due to the running, and hence the masses of the particles in the loop m i should be even smaller than m eff min (cf. Eq. (14)). In other words, if one adjusts the Yukawa couplings at TeV scale to larger values, the scale of the Landau pole of the Yukawa coupling becomes even smaller than those of the gauge couplings. (See the next section.)

Explicit examples
In the previous section, we have derived a generic upper bound on σ(pp → S → γγ) for given cutoff scale Λ and effective mass scale m eff min . Although it is a prominent implication of the diphoton resonance, m eff min does not directly correspond to physical masses of new charged particles. In order to see how light the charged particles should be in models with fermion loops, in this section we consider the running of the Yukawa couplings with concrete examples. We also briefly discuss the LHC constraints on vector-like quarks, and comment on the running of the scalar quartic coupling of S. In this section, we only consider the case of narrow width and take Γ S,total = V V =gg,γγ,γZ,ZZ,W W Γ(S → V V ).
For simplicity, we consider the N copies of Dirac fermions which transform as (R (3) , R (2) , Y ) under the SM gauge group, with universal Yukawa coupling and mass, y (5)i = y and m i = m. The RG equation for Yukawa coupling is given by [11] 16π 2 dy d ln µ = 3 + 2d (2) which holds both for scalar S (y = y i ) and pseudoscalar S (y = y 5i ). For a given representation (R (3) , R (2) , Y ), one can obtain the upper bound on σ(pp → S → γγ) as a function of m and Λ from the following procedure.
1. An upper bound on the multiplicity N , N max , is obtained as a function of m and Λ, by requiring that (i) the gauge couplings remain perturbative up to the scale Λ, and (ii) ∆b 3 satisfies the constraint from the α 3 (µ) measurement [8] (see Sec. 2). For the former constraint, we require α a (Λ) ≤ 1 for a = 1-3 in the numerical calculation.
2. For a given N (1 ≤ N ≤ N max ), an upper bound on the Yukawa coupling at low energy, y(µ = m), is obtained by requiring that the running Yukawa coupling, y(µ), also remains perturbative for µ < Λ. This gives the upper bounds on σ(pp → S → γγ) for a given set of (m, Λ, N ). Because y(m) increases as y(Λ) increases, we take y(Λ) = 4.
(We have checked that the maximal possible value of the cross section does not change much as far as y(Λ) is large enough.) 3. The maximum signal rate σ(pp → S → γγ) max is obtained with respect to N .
In the case where there is only one representation, N = N max gives the maximum value of σ(pp → S → γγ) with m, Λ, and the representation of the fermion being fixed. This is because the maximal value of the Yukawa coupling at low energy roughly scales as y ∼ N −1/2 , and therefore the signal rate increases as σ(pp → S → γγ) ∼ (N y) 2 ∼ N .
• Finally, in the case of SU(2) singlet (3, 1, 2/3), the signal rate is suppressed compared with the other two cases. In this case, the running α 3 constraint determines the N max in a large part of the low mass region m 600-700 GeV. The zigzag lines for m 600 GeV is due to the rapid increase of allowed N max (∆b 3 ) with respect to m from the running α 3 constraint. In each narrow range of m with a fixed N max , the upper bound on Λ is determined either by the perturbativity of the Yukawa coupling or by the Landau pole of U(1) Y .
Next, we discuss the constraint from the direct searches for vector-like quarks at the LHC. Here, we assume that they decay into the SM particles via a renormalizable coupling with SM quarks and the Higgs boson. In order to avoid the stringent constraint from the decay into third generation quarks, let us further assume that the coupling with the third generation is suppressed. Then, the vector-like quarks decay into a light SM quark and a W /Z/Higgs boson, depending on its representation. In particular, the search for a vectorlike quark decaying into a W boson and a light SM quark at the LHC gives a stringent constraint in the present scenario. From the result of ATLAS [12], the bound is estimated where Q and q denote the vector-like quark and the SM light quark, respectively.
• In the case of the SU(2) doublet (3, 2, 7/6), it contains vector-like quarks with electric charges of 5/3 and 2/3. The one with the electric charge of 5/3 decays into a W boson and a light SM quark (up and/or charm) with almost 100% branching fraction.
Comparing the bound in (20) with the lines in Fig. 2, if we require σ(pp → S → γγ) = 10 fb in the case of scalar S, the region of Λ 10 12 GeV (N = 1) is excluded, and Λ 10 9.5 GeV (N = 2) is at the boundary of excluded region. In the case of pseudoscalar, the model can explain σ(pp → S → γγ) = 10 fb while being perturbative up to Λ 10 17 GeV (Λ 10 12 GeV), if the vector-like quarks are as light as about 740 GeV (1100 GeV).
• In the case of SU(2) triplet (3, 3, 2/3), one of the SU(2) triplet quarks decays into W q with an almost 100% branching fraction, and another one has about 50% branching. Thus, from the bound (20), the region of m 750 GeV is excluded even for N = 1. The scalar case cannot have a cutoff larger than about 10 9 GeV in order to have the cross section larger than ∼ 3 fb, while the pseudoscalar case with N = 1, m 860 GeV, and σ(pp → S → γγ) = 5 fb is still allowed and can be perturbative up to the Planck scale.
• Finally, in the case of SU(2) singlet (3, 1, 2/3), the direct search excludes a large fraction of the parameter space with a sizable signal cross section, in particular when the cutoff scale is high. In this case, the vector-like quark decays into a W boson and a light quark with a branching fraction of about 50%. From the direct search bound in Eq. (20), the number of multiplicity should satisfy N < 2, 3, and 4 for m ≤ 690 GeV, m (690-750) GeV, and m (750-800) GeV, respectively. Thus, the lines in the figure for m 800 GeV are not consistent with the direct search bound. If we adopt the maximal number of multiplicity allowed by the direct search, the Yukawa coupling should be quite large at low energy in order to explain the diphoton signal. Even for the pseudoscalar case and for σ(pp → S → γγ) = 3 fb, the required value of the Yukawa coupling is y(m) 2.1, 1.5, and 1.2, for m ≤ 690 GeV, m (690-750) GeV, and m (750-800) GeV, respectively. If the RG equation (18) is evolved from low energy to high energy, they quickly become non-perturbative, which leads to cutoff scales below 10 TeV.
We should note that the above constraints strongly depend on the decay modes of vector-like quarks. If they mainly couple to the third generation SM quarks and decay into top and/or bottom quarks, the constraints become severer. Instead, if they decay in a very complicated way (e.g., in a cascade decay chain with multiple intermediate new particles emitting many soft jets), they may escape the direct search even for small mass region. Now let us briefly discuss the other representations in Eq. (19).
• In the case of (3, 2, 1/6), the pseudoscalar case can have σ(pp → S → γγ) = (5-10) fb with a large cutoff, but it requires a small mass m and a large multiplicity N . We found that the region below m 800 GeV is excluded if the vector-like quarks mainly decay into the SM light quarks, and for m 800 GeV the cutoff cannot be larger than 10 9 GeV for σ(pp → S → γγ) ≥ 5 fb.
Before closing this section, we comment on the running of the quartic coupling of the S field. 7 Defining the coupling λ as L S 4 = −(1/4!)λS 4 , its RG equation is given by We have checked that, as far as λ is positive at the cutoff scale, it does not become negative for m < µ < Λ and hence there is no vacuum instability. In addition, λ does not blow up below the cutoff scale irrespective of the value λ(Λ). Thus, there is no constraint from the running of the quartic coupling.

Conclusions
Motivated by the recent LHC results, we have studied the diphoton resonance production cross section at the LHC, paying particular attention to the running of the gauge and Yukawa coupling constants. We have considered the case where a (pseudo-)scalar particle S with its mass of 750 GeV is responsible for the diphoton events observed by the LHC and the scalar particle is produced by the gluon fusion. In such a case, new fermions and/or bosons which have SM gauge quantum numbers are necessary to generate S-g-g and S-γ-γ vertices.
Assuming that the S-g-g and S-γ-γ vertices are perturbatively generated by the loop effects of the new fermions and/or bosons, we studied how large the cross section for the process pp → S → γγ can be. We have shown that the cross section is severely constrained from above by (i) the perturbativity of the coupling constants up to a certain scale, and (ii) the consistency of the scale dependence of α 3 with that observed by the LHC. First, we have pointed out that a model-independent upper bound on σ(pp → S → γγ) can be derived, taking account of the two requirements mentioned above. Such a bound is obtained from the fact that the cross section is related to Γ(S → gg) and Γ(S → γγ), and that the amplitudes for these decay rates are proportional to the β-function coefficients of the gauge coupling constants from the fermions and bosons inside the loop. We have also calculated such a bound as a function of the cutoff scale Λ and the m eff min parameter which corresponds to the mass scale of the fermions and bosons inside the loop. (See Fig. 1.) Then, we have discussed the upper bound on σ(pp → S → γγ) in models with fermion loops, taking into account the perturbativity of the Yukawa coupling between S and the new fermions. For such a study, the particle content should be fixed to perform the RG analysis. We have considered seven possible representations of the fermions with which the fermions can directly decay into SM particles. We have introduced N copies of fermions in the same representation with the universal mass of m, and derived the upper bounds on σ(pp → S → γγ). Among them, the representation of (3, 2, 7/6) can give the largest diphoton rate in most of the parameter region. For instance, in the case of pseudoscalar, it is shown that σ(pp → S → γγ) = 5 and 10 fb can be obtained with m 1000 and 740 GeV (m 1600 and 1100 GeV) and N = 1 (N = 2) when the cutoff scale is 10 16 GeV (10 10 GeV), respectively. We have also discussed that such sets of parameters are consistent with the current constraints on vector-like quarks from the direct search at the LHC. In the cases of the other representations, the signal rate σ(pp → S → γγ) is more suppressed, and a large cutoff scale is impossible at all in some cases.
The present study suggests that, unless the cutoff scale is very low, there must exist new particles at TeV scale or lower. They should be an important target of the LHC run-2 and other future collider experiments.