Dispersive determination of fourth generation quark masses

We determine the masses of the sequential fourth generation quarks $b'$ and $t'$ in the extension of the Standard Model by solving the dispersion relations associated with the mixing between the neutral states $Q\bar q$ and $\bar Qq$, $Q$ ($q$) being a heavy (light) quark. The box diagrams responsible for the mixing, which provide the perturbative inputs to the dispersion relations, involve multiple intermediate channels, i.e., the $ut$ and $ct$ channels, $u$ ($c$, $t$) being an up (charm, top) quark, in the $b'$ case, and the $db'$, $sb'$ and $bb'$ ones, $d$ ($s$, $b$) being a down (strange, bottom) quark, in the $t'$ case. The common solutions for the above channels lead to the masses $m_{b'}=(2.7\pm 0.1)$ TeV and $m_{t'}\approx 200$ TeV unambiguously. We show that these superheavy quarks, forming bound states in a Yukawa potential, barely contribute to Higgs boson production via gluon fusion and decay to photon pairs, and bypass current experimental constraints. The mass of the $\bar b'b'$ ground state is estimated to be about 3.2 TeV. It is thus worthwhile to continue the search for $b'$ quarks or $\bar b'b'$ resonances at the (high-luminosity) large hadron collider.


I. INTRODUCTION
Our recent dispersive analyses of some representative physical observables (heavy meson decay widths, neutral meson mixing, etc.) have accumulated substantial indications that the scalar sector of the Standard Model (SM) is not completely free, but arranged properly to achieve internal dynamical consistency [1][2][3].Fermion masses can be derived by solving the dispersive relations for decay widths of a heavy quark Q as an inverse problem [4][5][6][7]: starting with massless final-state up and down quarks, we demonstrated that the solution for the Q → du d (Q → cūd) mode with the leading-order heavy-quark-expansion input yields the charm-quark (bottom-quark) mass m c = 1.35 (m b = 4.0) GeV [1].Requiring that the dispersion relation for the Q → su d (Q → dµ + ν µ , Q → uτ − ντ ) decay generates the identical heavy quark mass, we deduced the strange-quark (muon, τ lepton) mass m s = 0.12 GeV (m µ = 0.11 GeV, m τ = 2.0 GeV).The similar studies of fermion mixing [3] established the connections between the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and quark masses, and between the Pontecorvo-Maki-Nakagawa-Sakata matrix elements and neutrino masses.These connections explained the known numerical relation V us ≈ m s /m b [8], V us being a CKM matrix element, and the maximal mixing angle θ 23 ≈ 45 • in the lepton sector, and discriminated the normal hierarchy for neutrino masses from the inverted hierarchy or quasi-degenerate spectrum.
The dispersion relation for the correlation function of two b-quark scalar (vector) currents, with the perturbative input from the b quark loop, returns the Higgs (Z) boson mass 114 (90.8)GeV [2] in accordance with the measured values.It implies that the parameters µ 2 and λ in the Higgs potential are also constrained by internal dynamical consistency.Particle masses and mixing angles in the SM originate from the independent elements of the Yukawa matrices [9], as the electroweak symmetry is broken.Inspired by the above observations, we attempt to make a bold conjecture that the SM contains only three fundamental parameters actually, i.e., the three gauge couplings, and the other parameters, governing the interplay among various generations of fermions, are fixed by SM dynamics itself.The analyticity, which is inherent in quantum fields theories, imposes additional constraints.Its impact is not revealed in naive parameter counting at the Lagrangian level based on symmetries, but through dispersive analyses of dynamical processes.Dispersion relations, which physical observables like heavy-to-light decay widths must respect, link different types of interactions at arbitrary scales.The resultant constraints are so strong that the parameters in the scalar sector must take specific values, instead of being discretionary.
To maintain the simplicity and beauty conjectured above, a natural extension of the SM is to introduce the sequential fourth generation of fermions, since the associated parameters in the scalar sector are not free.That is, their masses and mixing with lighter generations can be predicted unambiguously in a similar manner [2].We first determine the top quark mass m t by solving the dispersion relations for the mixing between the neutral states Qū and Qu.The perturbative inputs to the dispersion relations come from the imaginary contributions of the box diagrams for the mixing with the intermediate db, sb and bb channels.Given the corresponding thresholds m d + m b , m s + m b , and 2m b for the typical quark masses m d = 0, m s = 0.1 GeV, and m b = (4.15± 0.01) GeV, we extract m t = (173 ± 3) GeV from the common solution to the three channels.The existence of such a common solution is highly nontrivial, making convincing our formalism and predictions obtained from it.We then go ahead to calculate the masses of the sequential fourth generation quarks b ′ and t ′ in the same framework, considering the multiple intermediate channels ut and ct in the b ′ case, and db ′ , sb ′ and bb ′ in the t ′ case.It will be observed that the common solutions for the various channels also exist, and demand the masses m b ′ = (2.7 ± 0.1) TeV and m t ′ ≈ 200 TeV.

𝑚𝑚 𝑚𝑚 𝑄𝑄
FIG. 1: Contour for the derivation of Eq. ( 1), where the thick lines represent the branch cuts.
Many merits of the sequential fourth generation model have been explored: condensates of the fourth generation quarks and leptons could be the responsible mechanism of the dynamical electroweak symmetry breaking [10,11]; electroweak baryogenesis through the first-order phase transition could be realized in this model [12]; it could provide a viable source of CP violation for the baryon asymmetry of the Universe based on the dimensional analysis of the Jarlskog invariants [13].However, it is widely conceded that this model has been ruled out mainly by the data of Higgs boson production via gluon fusion gg → H and decay into photon pairs H → γγ [14].Measurements of the oblique parameters, which depend on the additional mixing angles associated with the fourth generation quarks and the unclear contribution from the fourth generation leptons [15], give relatively weaker constraints.We point out that the superheavy fourth generation quarks b ′ and t ′ with the aforementioned masses form bound states in a Yukawa potential [16,17].Once they form bound states, physical degrees of freedom change, and new resonances emerge, so one has to reformulate the interaction between the fourth generation quarks and Higgs bosons with these new resonances [18].We will show that the b′ b ′ scalars contribute to the gg → H cross section only at 10 −3 level, relative to that from the top-quark loop in the SM.It is thus likely for the sequential fourth generation model to bypass the current experimental constraints, even without the expansion of the scalar sector [19].For an analogous reason, the model could also bypass the constraint from Higgs boson decay to photon pairs.The rest of the paper is organized as follows.We compute the top quark mass from the dispersion relations for the Qū and Qu mixing through the db, sb and bb channels in Sec.II.The framework is extended to the prediction for the b ′ (t ′ ) quark mass in Sec.III by investigating the multiple intermediate ut and ct (db ′ , sb ′ and bb ′ ) channels.The properties of the b′ b ′ scalar bound states S in a Yukawa potential, including the binding energies and the widths, are derived in Sec.IV, based on which we estimate the gg → S → H cross sections using the ggS and SH effective couplings and Breit-Wigner propagators for S. In particular, the mass of the b′ b ′ ground state, being either a pseudoscalar or a vector, is evaluated in a relativistic approach and found to be about 3.2 TeV.Some processes, which are promising for searching for b ′ quarks and their resonances at the (high-luminosity) large hadron collider, are proposed.Section V contains the summary.

II. FORMALISM AND TOP QUARK MASS
Consider the mixing between the neutral states Qū and Qu through the box diagrams with a heavy quark Q of mass m Q and a massless u quark [2,20].The construction of a dispersion relation follows the procedure in [1] straightforwardly, which starts with the contour integration of the mixing amplitudes Π ij , ij = db, sb and bb, in the complex m plane.The contour consists of two pieces of horizontal lines above and below the branch cut along the positive real axis, two pieces of horizontal lines above and below the branch cut along the negative real axis, a small circle around the pole m = m Q located on the positive real axis and a circle C R of large radius R as depicted in Fig. 1.As recollected in Appendix A, we have the dispersion relations for the imaginary pieces of ( The quark-level thresholds m ij for the box-diagram contributions ImΠ box ij denote m i + m j , i.e., m db = m d + m b , m sb = m s + m b and m bb = 2m b .The physical quantities ImΠ ij (m) on the left-hand side of the above expression have the hadronic thresholds M db = m π + m B , M sb = m K + m B and M bb = 2m B with the pion (kaon, B meson) mass m π (m K , m B ).The CKM factors associated with the db, sb, and bb channels can vary independently in a mathematical viewpoint, so their corresponding dispersion relations can be analyzed separately.These dispersion relations, holding for arbitrary m Q , impose stringent connections between high-mass and low-mass behaviors of the mixing amplitudes.
The box diagrams generate two effective four-fermion operators of the (V −A)(V −A) and (S−P )(S−P ) structures.Viewing that the two structures endow separate dispersion relations, and the latter also receives contributions from amplitudes other than the box diagrams, like the double penguin amplitude [21], we concentrate on the former.The imaginary piece of the (V − A)(V − A) structure in perturbative evaluations [22,23] is written as with the W boson mass m W and the intermediate quark masses m i and m j .A d quark is also treated as a massless particle, i.e., m d = 0.The overall coefficient, irrelevant to the derivation below, is implicit.We have kept only the Wilson coefficient C 2 (µ) [24], which dominates over C 1 (µ) at the renormalization scale µ = m Q ≥ m b .The second term in the curly brackets of Eq. ( 2) is down by a tiny ratio (m 2 i + m 2 j )/m 2 W , so the behavior of Eq. ( 2) in m Q is dictated by the first term.In the threshold regions with m Q ∼ m ij , it is approximated by Because of m s ≪ m b , m b − m s is not very distinct from m b + m s , and the dependence on the former has been retained in the second line of Eq. (3).Motivated by the above threshold behaviors, we choose the integrands for the dispersion integrals in Eq. (1) as [2] ImΠ where Γ ij (m) are the unknowns to be solved for shortly, and the definitions of ImΠ box ij (m) by means of Γ box ij (m) should be self-evident.Note that Γ box bb (m) is an odd function in m, which accounts for the odd power of m in the numerator of ImΠ bb (m) [2].The above integrands with powers of m in the numerators suppress any residues in the low m region, including those from the poles at m = ±(m i + m j ) and m = ±(m i − m j ), compared to the ones from Moving the integrands on the right-hand side of Eq. ( 1) to the left-hand side, we arrive at with the subtracted unknown functions ∆ρ ij (m Owing to the subtraction of the boxdiagram contributions and the limits ImΠ ij (m) → ImΠ box ij (m) at large m, the integrals in Eq. ( 5) converge even after the upper bound of m 2 is extended to infinity.The unknowns ∆ρ ij (m) are fixed to the initial conditions −ImΠ box ij (m) in the interval (m ij , M ij ) of m, in which the physical quantities ImΠ ij (m) vanish.The idea behind our formalism is similar to that of QCD sum rules [25], but with power corrections in (M ij − m ij )/m Q arising from the difference between the quark-level and hadronic thresholds, which are necessary for establishing a physical solution [20].As seen later, it is easier to solve for ∆ρ ij (m because the initial conditions of the former are simpler.Once ∆ρ ij (m Q ) are attained, we convert them to ∆Γ ij (m Q ) following Eq.( 4).Without the power corrections, i.e., if m ij are equal to M ij , there will be only the trivial solutions ) and no constraint on the top quark mass.The steps of solving Eq. ( 5) have been elucidated in [1] and briefly reviewed in Appendix A. The solution of the unknown function can be constructed with a single Bessel function of the first kind J α (x), A solution to the dispersion relation must not be sensitive to the arbitrary scale ω, which results from scaling the integration variable m 2 in Eq. ( 5) artificially [2].To realize this insensitivity, we make a Taylor expansion of ∆ρ ij (m Q ), where the constant ωij , together with the index α ij and the coefficient y ij , are fixed through the fit of the first term ∆ρ ij (m Q )| ω=ωij to the initial condition in the interval (m ij , M ij ) of m Q .The insensitivity to the variable ω commands the vanishing of the first derivative in Eq. ( 7), d∆ρ ij (m Q )/dω| ω=ωij = 0, from which roots of m Q are solved.Furthermore, the second derivative d 2 ∆ρ ij (m Q )/dω 2 | ω=ωij should be minimal to maximize the stability window around ωij , in which ∆ρ ij (m Q ) remains almost independent of ω.
The threshold behaviors in Eq. ( 3) and the initial conditions The solution in Eq. ( 6) scales in the the threshold region ] αij owing to the relation J α (z) ∼ z α in the limit z → 0. Contrasting this scaling law with Eq. ( 8), we read off the indices It is clear now why we employed those modified integrands in Eq. ( 4); the corresponding inputs in Eq. ( 8) are proportional to simple powers of m 2 Q − (m i + m j ) 2 , so that the indices α ij can be specified unambiguously.The coefficients y ij are related to the boundary conditions at the high end , which fix the coefficients The running coupling constant is given by with the coefficients β 0 = 11 − 2n f /3 and β 1 = 2(51 − 19n f /3).We take the QCD scale Λ QCD = 0.21 GeV for the number of active quark flavors n f = 5 [26], and choose the renormalization scale µ = m Q as stated before.Note that we need only the quark-mass inputs for the initial conditions in the interval (m ij , M ij ) of m Q .Adopting the quark masses m s = 0.1 GeV and m b = 4.15 GeV in the MS scheme at the scale µ ∼ m b , which are close to those from lattice calculations [28], and the pion (kaon, B meson) mass m π = 0.14 GeV (m K = 0.49 GeV, m B = 5.28 GeV) [27], we get ωdb = 0.0531 GeV −1 , ωsb = 0.0268 GeV −1 ωbb = 0.0128 GeV −1 from the best fits of ∆ρ ij (m Q ) in Eq. ( 6) to The fit results by means of ∆Γ ij (m Q ), which are related to ∆ρ ij (m Q ) via Eq.( 4), are compared with −Γ box ij (m Q ) in the interval (m ij , M ij ) in Fig. 2. Their perfect matches confirm that the approximate solutions in Eq. ( 6) work well, and that other methods for obtaining ωij should return similar values.For example, equating ∆ρ ij (m Q ) and /2 leads to ωdb = 0.0503 GeV −1 , ωsb = 0.0268 GeV −1 and ωbb = 0.0129 GeV −1 , very close to those from the best fits.
The unknown subtracted functions ∆ρ ij (m Q ) with the above α ij , y ij and ωij are displayed in Fig. 3(a) through ∆Γ ij (m Q ).They exhibit oscillatory behaviors in m Q , and the first (second, third) peak of the solution for the bb (sb, db) channel is located around m Q ≈ 170-195 GeV.The coincidence between the sequences of the peaks and of the quark generations is intriguing.The similar feature will appear again in the plots for the fourth generation quark masses in the next section.To evince the implication of the above peak overlap, we present in Fig. 3 where the factors independent of ω have been dropped for simplicity.
The band of the bb curve is induced by the variation of the bottom quark mass m b in the range m b = (4.15± 0.01) GeV with roughly 1σ deviation from the value 4.18 +0.03 −0.02 GeV in [27].The considered error of m b is also compatible with that obtained in Ref. [28].The result for the db channel is less sensitive to m b , but depends more strongly on the methods of determining ωdb as mentioned before.Namely, the band of the db curve is mainly attributed to the latter source of uncertainties with ωdb being lowered to 0.0503 GeV −1 .The derivative D sb (m Q ) is stable with respect to various sources of uncertainties; for instance, changing the strange-quark mass m s by 10% causes only about 1% effects.It is the reason why the sb curve discloses a narrow band.Every curve in Fig. 3(b) indicates the existence of multiple roots.It has been checked that the second derivatives are larger at higher roots [1], so smaller roots are preferred in the viewpoint of maximizing the stability windows in ω. Figure 3(b) shows that the three derivatives first vanish simultaneously around m Q ≈ 173 GeV, as manifested by the intersection of the three curves in the interval (170 GeV, 176 GeV), which corresponds to the location of the peak overlap in Fig. 3(a).To be explicit, we read off the roots m Q = 169.1 +9.5 −1.1 GeV for the db channel, m Q = 176.2± 0.6 GeV for the sb channel and m Q = 175.7 +7.3 GeV for the bb channel in Fig. 3(b).The result of m Q , as a common solution to the considered channels, is identified as the physical top quark mass, which agrees well with the observed one m t = (172.69± 0.30) GeV [27].
A remark is in order.The tiny error 0.01 GeV for the input m b = (4.15± 0.01) GeV was adopted to examine the sensitivity of our predictions to the variation of the bottom quark mass.We emphasize that the main purpose of the present work is to predict the fourth generation quark masses, for which both the bottom and top quark masses are necessary inputs.Hence, the reproduction of the top quark mass from the given bottom quark mass in its allowed range is not only to validate our formalism, but to calibrate the inputs for the predictions.This calibration is essential owing to the sensitivity to the inputs as noticed above (the determination of the lighter quark masses in our formalism is more stable against variations of inputs [1]).Besides, we set the renormalization scale to the invariant mass m Q of the heavy quark in Eq. ( 2), and stick to this choice for the consistent determination of the top quark mass and the fourth generation quark masses.We think that m b = 4.15 ± 0.01 GeV and the resultant m t = 173 ± 3 GeV, in agreement with the extractions from other known means and current data, serve as the appropriate inputs.Note that only the outcome from the bb channel, which involves two bottom quarks in the intermediate states, is sensitive to the input of m b .Therefore, a resolution to the aforementioned sensitivity that one can make is to discard the bb channel, and to consider simply the db and sb channels.The simultaneous vanishing of their derivatives in Eq. ( 12) is sufficient for deriving a stable and definite top quark mass.

III. FOURTH GENERATION QUARK MASSES
After verifying that the dispersive analysis produces the correct top quark mass, we extend it to the predictions of the fourth generation quark masses, starting with the b ′ one.Consider the box diagrams for the mixing of the neutral states Q d and Qd, and construct the associated dispersion relations.The intermediate channels, which contribute to the imaginary pieces of the box diagrams, contain not only those from on-shell quarks ut, ct and tt described by Eq. ( 2), but those from on-shell W bosons.Since these channels can be differentiated experimentally, we can focus on the former for our purpose.The necessary power corrections proportional to the differences between the quark-level thresholds m ij and the physical thresholds M ij further select the ut channel with m ut = m t (m u = 0) and M ut = m π + m t , and the ct channel with m ct = m c + m t and M ct = m D + m t , m D being the D meson mass.Note that the second term in the curly brackets of Eq. ( 2) becomes more important in the present case owing to the large ratio (m Because of m c ≪ m t , the terms (m t − m c ) 2 and m 2 t + m 2 c , which are not very distinct from (m t + m c ) 2 , have stayed in the second line of Eq. (13).
Motivated by the above threshold behaviors, we choose the integrands for the dispersion integrals in Eq. ( 1) as Γ (GeV )  6) and ( 10), respectively.The initial conditions near the thresholds m Q ∼ m ij are given by which assign the indices For the numerical study, we take the QCD scale Λ QCD = 0.11 GeV for the number of active quark flavors n f = 6 according to [29] Λ with m t = 173 GeV and Λ (5) QCD = 0.21 GeV [26].The behaviors of the box-diagram contributions Γ box ij (m Q ) in the interval (m ij , M ij ) of m Q matter in solving the dispersion relations.In view of the high top-quark mass, the renormalization-group (RG) evolution of the charm-quark mass to a scale of O(m t ) needs to be taken into account.This RG effect is minor in the previous section, since m b does not deviate much from the range µ ≈ 1-2 GeV, in which the strange-quark mass is defined.We have at µ = m t [26] for m c (m c ) = 1.35 GeV [1].The inputs of the pion mass m π = 0.14 GeV and the D-meson mass m D = 1.87 GeV [27] then yield ωut = 0.00326 GeV −1 and ωct = 0.00176 GeV −1 from the best fits of Eq. ( 6) to )/2 generates ωut = 0.00326 GeV −1 and ωct = 0.00175 GeV −1 , almost identical to the values from the best fits.This consistency supports the goodness of our solutions.
The dependencies of the unknown subtracted functions ∆ρ ij (m Q ) on m Q from solving the dispersion relations are presented in Fig. 4(a) by means of ∆Γ ij (m Q ).We have confirmed the excellent matches between ∆Γ ij (m Q ) form the fits and the initial conditions feature noticed before hints that the second (third) peak of the curve for the ct (ut) channel should be located at roughly the same m Q .Figure 4(a), with the overlap of peaks around m Q ≈ 2.7 TeV, corroborates this expectation.The corresponding derivatives in Eq. ( 12) as functions of m Q are drawn in Fig. 4 respectively.Since a top quark does not form a hadronic bound state, we do not expect that a b ′ quark will, and keep the quark mass m b ′ in the hadronic thresholds.Certainly, this is an assumption owing to the the uncertain 4 × 4 CKM matrix element V tb ′ .The second term in the curly brackets of Eq. ( 2) dominates because of the large ratio (m The behaviors of Eq. ( 2) in the threshold regions with m Q ∼ m ij are approximated by Eq. ( 13), with the first line for the db ′ channel and the second line for the sb ′ and bb ′ channels.The appropriate replacements of the masses m c,t by m s,b,b ′ are understood.The modified integrands for the dispersion integrals in Eq. ( 1) and their expressions near the thresholds m Q ∼ m ij follow Eqs.( 14) and (15), respectively, also with the first lines for the db ′ channel and the second lines for the sb ′ and bb ′ channels.We then acquire the indices The QCD scale takes the value Λ QCD = 0.04 GeV for n f = 7 according to Eq. ( 17) but with m b ′ being substituted for m t .The RG effects on the quark masses are included via Eq.( 18), which give m s ≈ 0.07 GeV and m b ≈ 3.2 GeV at the scale µ = m b ′ .Inputting the same masses m π , m K , m B and m b ′ = 2.7 TeV, we get ωdb ′ = 0.0438 TeV −1 , ωsb ′ = 0.0223 TeV −1 , and ωbb ′ = 0.0233 TeV −1 from the best fits of Eq. ( 6 5(a) collects the solutions ∆Γ ij (m Q ) as functions of m Q , where the curves for the bb ′ and sb ′ channels are close in shape, and their second peaks overlap with the third peak for the db ′ channel around m Q ≈ 200 TeV.The bb ′ and sb ′ channels share the identical formula characterized by the same indices α sb ′ = α bb ′ = −1/2.Moreover, the difference between m s and m b (also between m K and m B ) is minor relative to the high m b ′ , so that these two solutions behave similarly.Hence, there are only two categories of solutions in the t ′ case, and the overlap takes place between the second and third peaks.
The dependencies of the derivatives D ij (m Q ) on m Q , defined in Eq. ( 12), are displayed in Fig. 5(b) for ω = ωij .The three derivatives first vanish simultaneously around m Q ≈ 200 TeV, which coincides with the aforementioned peak locations.It is sure that a common root for the fourth generation quark mass m t ′ exists.Since the value of m t ′ is obviously beyond the current and future reach of new particle searches, we do not bother to include theoretical uncertainties with the prediction.One may wonder whether m t ′ ≈ 200 TeV violates the unitarity limit signified by the large Yukawa coupling.However, bound states would be formed in this case, such that physical degrees of freedom change, and the high Yukawa coupling is not an issue.This subject will be elaborated in the next section.It is not unexpected that a t ′ quark is so heavy, viewing that a c quark is 13 times heavier than an s quark, and a t quark is about 40 times heavier than a b quark.Here a t ′ quark is about 70 times heavier than a b ′ quark.

IV. b′ b ′ BOUND STATES
As remarked in the Introduction, the sequential fourth generation model is disfavored by the data of Higgs boson production via gluon fusion and decay into photon pairs [14].Nevertheless, it has been known [16] that the fourth generation quarks, whose mass m Q meets the criterion 68, with the vacuum expectation value v = 246 GeV, form bound states in a Yukawa potential.The binding energy for the QQ ground state with the masses m * Q ≈ 1.26 TeV and m * H ≈ 1.45 TeV at the fixed point of the RG evolution in this model was found to be −4.9GeV.The fixed point depends on the initial values of the quark masses at the electroweak scale of O(100) GeV: the larger the initial values, the lower the fixed point is.The b ′ quark mass m b ′ = 2.7 TeV predicted in the previous section, greater than the fixed-point value 1.26 TeV, satisfies the criterion K Q > 1.68 definitely.The binding energy for the b′ b ′ bound state ought to be higher.We will demonstrate that the new scalars S formed by b′ b ′ , with tiny couplings to a Higgs boson, escape the current experimental constraints.It is then worthwhile to keep searching for a superheavy b ′ quark at future colliders [30].
Once the bound state of mass at TeV scale is formed, the gluon fusion process involving internal b ′ quarks at the low scale m H should be analyzed in an effective theory with different physical degrees of freedom.In other words, one has to regard the process as gluon fusion into the scalar S, followed by production of a Higgs boson through a coupling between them.The order of magnitude of the corresponding amplitude is assessed below.First, the gluon fusion into S is proportional to √ sg ggS , where the invariant mass √ s of S takes into account the dimension of the effective operator A µ A ν S, A µ being a gluon field, and g ggS is a dimensionless effective coupling.The scalar S then propagates according to a Breit-Wigner factor 1/(s − m 2 S − i √ sΓ S ), where Γ S denotes the width of S. At last, S transforms into a Higgs boson H with the magnitude being described by sg SH , where g SH is a dimensionless effective coupling.The total amplitude is thus written, in the effective approach, as where factors irrelevant to our reasoning have been suppressed.Properties of heavy quarkonium states, like b′ b ′ , in a Yuakawa potential of the strength α Y = m 2 b ′ /(4πv 2 ), have been explored intensively in the literature (for a recent reference, see [31]).With the involved superheavy quark mass scale, we have adopted the fixed-point Higgs boson mass m * H in the exponential.Note that the number of bound states is finite for a Yukawa potential, distinct from the case for a Coulomb potential which allows infinitely many bound-state solutions.It turns out that only the states characterized by (n, l) = (1, 0), (2, 0), (2, 1), (3, 0) and (3, 1) are bounded, n (l) being the principal (angular momentum) quantum number.The states labeled by (n, l) = (3, 2) or higher quantum numbers are not bounded.The ground state with (n, l) = (1, 0), being either a pseudoscalar or a vector, is expected to have a negligible coupling to a Higgs boson.It is easy to read off the value ϵ 10 = −0.75 of this state from Fig. 1 in [31], i.e., the binding energy E 10 ≡ α 2 Y m b ′ ϵ 10 /4 ≈ −41 TeV, for the parameter 1/(m * H a 0 ) = 8.9, a 0 ≡ 2/(α Y m b ′ ) being the Bohr radius.It is apparent that this deep ground state has revealed the nonrelativistic Thomas collapse [32], and calls for a relativistic treatment [17,33].
To examine the coupling to a Higgs boson, we concentrate on the P -wave scalar states with l = 1, and deduce the value ϵ 21 = −0.08 for the (n, l) = (2, 1) state from Fig. 2 in [31], i.e., the binding energy TeV.We suspect that this deep bound state also suffers from the Thomas collapse, but continue our order-of-magnitude estimate for completeness.Figure 5 in [31] provides the first derivative of the corresponding wave function at the origin for the parameter δ = m * H a 0 = 0.11.The width Γ S is then approximated by the S → gg decay width as in the heavy quarkonium case [34], where the strong coupling has been evaluated at twice of the b ′ quark mass, R ′ nl (0) = 4π/3ψ ′ nl (0) is the derivative of the radial wave function at the origin [31], and the scalar has the mass m S = 2m b ′ + E 21 ≈ 440 GeV.The width in Eq. ( 23) is larger than the scalar mass, signaling another warning to the consistency of this state.
To pin down the product of the effective couplings g ggS g SH , we match the amplitude in Eq. ( 20) to the one in the fundamental theory by considering the production of a fictitious Higgs boson with mass squared s ≈ m 2 S .The involved scale is so high, that the evaluation in the fundamental theory [35,36] based on the direct annihilation of the b′ b ′ quark pair ought to yield a result the same as in the effective approach.We identify the part of the amplitude, which approaches 3/2 in the lowest-order expression from the fundamental theory [35,36], The factor s − m 2 S ≪ √ sΓ S has been ignored in the denominator for s ≈ m 2 S on the right-hand side of the first equal sign.Equation ( 24) implies g ggS g SH = (2/3)Γ S /v obviously.We then obtain, by extrapolating Eq. ( 20) to s = m 2 H , the suppression factor on the S contribution relative to the top-quark one in the SM, The above result also suggests that the S contribution decreases like m −4 S .We repeat the discussion for the (n, l) = (3, 1) state, whose binding energy and the first derivative of the corresponding wave function at the origin read with ϵ 31 = −0.002according to Figs. 2 and 5 in [31], respectively.The width Γ S in Eq. ( 23) is given, for this state, by with m S = 2m b ′ + E 31 ≈ 5.28 TeV.The similar matching procedure leads to the diminishing suppression factor 2 3 on the S contribution to the Higgs boson production via gluon fusion in the SM.We confront the above estimates with those from the relativistic calculation [33], whose Eq. ( 28) indeed allows only the bound-state solutions characterized by n = 1, 2 and 3.Because of their crude approximation, the states labeled by the same n but different l are degenerate in eigenenergies.We take the positive eigenenergy E n from Eq. ( 28) of [33], extract the binding energy E b n = E n − m b ′ /2 with m b ′ /2 being the reduced mass of the b′ b ′ system, and derive the bound-state mass m S = 2m b ′ + E b n .It is trivial to get the ground-state mass 3.23 TeV, the mass of the first excited state 4.45 TeV for n = 2 and the mass of the second excited state ≲ 5.40 TeV for n = 3.The last value, differing from the nonrelativistic one 5.28 TeV by only 2%, confirms that this state is loosely bound.The masses of the first two states from the relativistic framework look more reasonable.We mention that a recent study of the oblique parameters S and T has permitted heavy resonances to be heavier than 3 TeV [37].Equations ( 23) and (27) hint that the widths of these bound states are of the same order of magnitude, so Eq. ( 25) indicates a tiny contribution from the n = 2 state at 10 −3 level to the Higgs boson production via gluon fusion.We conclude that the S contributions are negligible compared with the SM one.It is thus likely that a fourth generation quark as heavy as 2.7 TeV bypasses the constraint of the measured gg → H cross section at the scale s ∼ m 2 H .The same observation holds for the constraint on the fourth generation quarks from the data of the Higgs decay into photon pairs.The reasoning related to the H → γγ decay proceeds in a similar way.One just replaces the effective coupling g ggS in Eq. ( 24) by g γγS , and the constant 3/2 on the right-hand side of Eq. ( 24) by 1/2, which takes into account the color factor for the quark loop and the electric charge of a top quark.We then estimate the suppression factor on the S contribution relative to the top-quark one, That is, the contribution from the b′ b ′ bound state to the H → γγ decay is also negligible.It is impossible to detect a t ′ quark with a mass as high as 200 TeV in the foreseeable future.To detect a b ′ quark, the gluon fusion into a b′ b ′ resonance of mass about 3.2 TeV may not be efficient owing to the small gluon distribution functions at large parton momenta.Instead, the fusion process qq → W W, ZZ → S [38] is more promising, whose cross section is enhanced by the quark distribution functions.Another promising channel is the W -boson mediated single b ′ quark production associated with a top quark and a light quark, such as dg → u tb ′ .It gains the power enhancement with one fewer virtual weak boson by paying the price of having a smaller gluon distribution function.Presuming that b ′ decays into tW dominantly, one can search for an excess of t tW final states [39,40].The analysis is analogous to the search of vector-like heavy quarks [41], and the currently available strategies work.Another simpler single b ′ quark production from the ug → W + b ′ process may be attempted, which, however, suffers the uncertain suppression of the diminishing 4 × 4 CKM matrix element V ub ′ .

V. CONCLUSION
After accumulating sufficient clues in our previous studies that the scalar sector of the SM can be stringently constrained and there might be only three fundamental parameters from the gauge groups, we delved into the sequential fourth generation model as a natural extension of the SM.It has been demonstrated that the fourth generation quark masses can be predicted in the dispersive analyses of neutral quark state mixing involving a heavy quark.The idea is to treat the dispersion relations obeyed by the mixing observables as inverse problems with the initial conditions from the box-diagram contributions in the interval between the quark-level and hadronic thresholds.A heavy quark must take a specific mass in order to ensure a physical solution for the mixing observable to be invariant under the arbitrary scaling of the heavy quark mass in the dispersive integrals.We first worked on the mixing mediated by the db, sb and bb channels, and showed that the roots of the heavy-quark mass m Q corresponding to the first (second, third) peaks of the bb (sb, db) contributions, with the inputs of the typical strange-and bottom-quark masses, coincide around m Q ≈ 173 GeV.This outcome, highly nontrivial from the three independent channels and in agreement with the measured top quark mass, affirms our claim that the scalar interaction introduced to couple different generations in the SM is not discretionary.
Encouraged by the successful explanation of the top quark mass, we applied the formalism to the predictions for the fourth generation quark masses.The perturbative inputs to the dispersion relations come from the same box diagrams involving multiple intermediate channels, i.e., the ut and ct channels in the b ′ case, and the db ′ , sb ′ and bb ′ ones in the t ′ case.As expected, we solved for the common masses m b ′ = (2.7 ± 0.1) TeV and m t ′ ≈ 200 TeV from the above channels, which should be solid and convincing.Such superheavy quarks with the huge Yukawa couplings form bound states.The contributions from the b′ b ′ scalars to Higgs boson production via gluon fusion were assessed in an effective approach.Employing the eigenfunctions for scalar bound states in a Yukawa potential available in the literature, we calculated the widths appearing in the Breit-Wigner propagator associated with the scalars.We further fixed the relevant effective couplings for the gluon-gluon-scalar vertices and for the new scalar transition to a Higgs boson.The new scalar contributions at the scale of the Higgs boson mass turned out to be of O(10 −3 ) of the top-quark one in the SM at most, and is negligible.This estimate illustrated why these superheavy quarks could bypass the current experimental constraints from Higgs boson production via gluon fusion and decay to photon pairs, and why one should continue the search for fourth generation b ′ quarks or their resonances at the (high-luminosity) large hadron collider.
where N will be extended to infinity eventually.The first N generalized Laguerre polynomials L ij .We are allowed to treat ω as a finite variable, though both N and Λ can be arbitrarily large.The arbitrariness of Λ, which traces back to that of the large circle radius R, goes into the variable ω.The exponential suppression factor e −(m 2 −m 2 ij )/(2Λ) = e −ω 2 (m 2 −m 2 ij )/(2N ) is further replaced by unity for finite ω and large N .Equation (A11) then gives the solution in Eq. ( 6).

m
= ±m Q at large m Q .The denominators alleviate the divergent behaviors caused by the modified numerators at large m.The factor m 2 − (m b − m s ) 2 in Π sb (m) introduces an additional branch cut along the real axis in the interval −(m b − m s ) < m < m b − m s in the m plane.Our contour crosses the real axis between m = −(m b + m s ) and m = −(m b − m s ) and between m = m b − m s and m = m b + m s , and runs along the real axis marked by m < −(m b + m s ) and m > m b + m s , such that this additional branch cut does not contribute.

FIG. 4 :
FIG. 4: (a) Dependencies of the solutions ∆Γij(mQ) on mQ for ij = ut (solid line) and ij = ct (scaled by a factor 0.02, dashed line).(b) Dependencies of the derivatives Dij(mQ) on mQ.The curve for ij = ut has been scaled by a factor 0.01.
(b).Similarly, our results for the ct channel are insensitive to the variation of m c : 10% changes of m c stimulates only about 1% effects on the outcome of the fourth generation quark mass m b ′ .The uncertainties from different ways of fixing ωij are negligible as investigated above.Hence, we consider only the uncertainties from the variation of the top-quark mass within m t = (173 ± 3) GeV attained in the previous section, which are reflected by the bands of the curves.It is found that the two derivatives first vanish simultaneously around m Q ≈ 2.7 TeV, coinciding with the location of the peak overlap in Fig.4(a).That is, a common solution m b ′ = (2.7 ± 0.1) TeV, as inferred from Fig. 4(b), exists for the two considered channels.The prediction of the fourth generation quark mass m t ′ proceeds in exactly the same manner.The box diagrams governing the mixing of the neutral states Qū and Qu involve the intermediate db ′ , sb ′ and bb ′ channels, which are associated with the quark-level thresholds m db ′ = m b ′ (m d = 0), m sb ′ = m s + m b ′ and m bb ′ = m b + m b ′ , and the physical thresholds