PHYSICAL REVIEW D 104, 035031 (2021)

Implications of gravitational waves for supersymmetric grand unification

So Chigusa ,1,2,3 Yuichiro Nakai,4 and Jiaming Zheng4 1Berkeley Center for Theoretical Physics, Department of Physics, University of California, Berkeley, California 94720, USA 2Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan 4Tsung-Dao Lee Institute and School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240 China

(Received 22 November 2020; accepted 26 July 2021; published 27 August 2021)

Supersymmetric grand unification based on SOð10Þ is one of the most attractive paradigms in physics beyond the Standard Model. Inspired by the recent NANOGrav signal, we discuss the implications of detecting a stochastic gravitational wave background emitted by a network of cosmic strings for the SOð10Þ grand unification. Starting from a minimal model with multiple steps of symmetry breaking, we show that it generally prefers a high intermediate scale above 1014 GeV that is favored by observable primordial gravitational waves. The observed spectrum can potentially narrow the possible range of the cosmic string scale and restricts the unified couplings and the unification scale by requiring gauge coupling unification. As an indirect consequence of the high cosmic string scale, the monopole abundance places nontrivial constraints on the theory. These are complementary to the proton decay constraints and probe different facets of supersymmetric SOð10Þ unification theories.

DOI: 10.1103/PhysRevD.104.035031

I. INTRODUCTION the dark matter [11,12]. Therefore, the supersymmetric SOð10Þ grand unified theory (GUT) is one of the most The structure of the Standard Model (SM) matter sector, compelling frameworks of physics beyond the SM. quarks and leptons, and the high-energy behavior of the SM Since the grand unification is realized at a very high gauge couplings strongly suggest that the three SM gauge energy scale, its experimental test must be indirect. The groups G ≡ SUð3Þ × SUð2Þ × Uð1Þ are unified at a SM C L Y most famous prediction of the GUT is the finite lifetime of high-energy scale [1]. The smallness of neutrino masses the proton. The current lower limit from the Super- further indicates the existence of heavy right-handed Kamiokande experiment is 1.6 × 1034 years [13] for the neutrinos to realize the seesaw mechanism [2–4], which p → π0eþ decay mode and 5.9 × 1033 years [14] for the is a consequence of the grand unification based on p → Kþν¯ SO 10 [5,6]. A large hierarchy of scales between the mode. In the near future, the limit is expected to ð Þ 7 8 1034 3 2 1034 electroweak symmetry breaking (EWSB) and the grand be improved to . × years ( . × years) for the π0 þ þν¯ unification is naturally stabilized by supersymmetry e (K ) mode at the Hyper-Kamiokande experiment → þν¯ (SUSY). Amazingly, in the minimal supersymmetric SM [15]. Comparable sensitivities to the p K decay of 6 1034 1 9 1034 (MSSM), the precise unification of the three SM gauge × years and . × years are expected at the couplings is achieved. Moreover, with appropriate choices DUNE [16] and JUNO [17] experiments, respectively, with of Higgs representations to break the SOð10Þ,aZ2 somewhat different time scales. Another hint may come subgroup known as the matter parity remains unbroken from topological defects associated with GUT phase at all scales [7–10], while such parity is an ad hoc global transitions. In particular, a network of cosmic strings symmetry in the MSSM. The matter parity forbids the rapid can be produced and the matter parity renders them stable – decay of protons and ensures the stability of the lightest [18 20]. The cosmic strings form closed loops, shrink and supersymmetric particle which gives a viable candidate of lose energy via the emission of gravitational waves (GWs) [21,22].1 Interestingly, a stochastic GW background

Published by the American Physical Society under the terms of 1Numerical simulations based on Nambu-Goto strings support the Creative Commons Attribution 4.0 International license. this picture [23,24]. In the Abelian Higgs model, however, cosmic Further distribution of this work must maintain attribution to strings lose energy via particle productions, which suppress the the author(s) and the published article’s title, journal citation, GW production [25,26]. Recent simulations indicate that such and DOI. Funded by SCOAP3. particle productions are important only for very small loops [27].

2470-0010=2021=104(3)=035031(11) 035031-1 Published by the American Physical Society CHIGUSA, NAKAI, and ZHENG PHYS. REV. D 104, 035031 (2021) produced by cosmic strings stretches across a wide range of observation of cosmic string GW does not only imply the frequencies, and such a GW signal is one of the existence of an intermediate scale but also strongly moti- main targets in multifrequency GW astronomy and vates supersymmetric SOð10Þ as the unified theory of cosmology. gauge interactions. Recently, the NANOGrav collaboration has reported The remainder of this paper is organized as follows. In the first evidence of a stochastic GW background in pulsar Sec. II, we summarize the models and patterns of gauge timing data [28].2 Although the signal is not conclusive, it symmetry breaking considered in this paper. In Sec. III,we can be well fitted by GWs from a network of cosmic show the calculation of the running of gauge coupling strings because they can give a favored flat spectrum of constants and the resulting constraint from the gauge frequencies in the GW energy density [32–35].3 The coupling unification. Section IV and Sec. V are devoted energy density spectrum is proportional to ðGμÞ2 where to the calculation of the proton decay rate and the monopole G is the Newton’s constant and μ is the cosmic-string density, respectively, with showing the resulting con- tension. The spectrum also depends on the loop size at the straints. We conclude and give some discussions of our time of formation α. This parameter has not been reliably results in Sec. VI. estimated so far, but larger loop size α ¼ 0.01–0.1 is typically favored by recent numerical simulations II. MODELS [32,48,49]. Conservatively, taking α ¼ 3×10−4;5×10−3; 1×10−1, the NANOGrav signal can be fitted with SOð10Þ is a rank 5 simple group and contains an −7 −9 −10 1 Gμ ¼ 1 × 10 ; 5 × 10 ; 1 × 10 , respectively [33]. additional Uð Þ factor in addition to GSM of rank 4. The The symmetry breaking scale associated with the minimal choices of Higgs representations to break the Uð1Þ formation of cosmic strings is related to the string in a SUSY model are 16 þ 16 or 126 þ 126. We will tension, v ∼ 1016 GeVðGμ=10−7Þ1=2, whose precise co- stick to the latter choice as it preserves the matter parity 3 − efficient is given by an Oð1Þ group theory factor. Then, the M ≡ ð−1Þ ðB LÞ where B and L are baryon and lepton NANOGrav signal indicates the symmetry breaking scale numbers. Vacuum expectation values (VEVs) of SM is in the range, 1014 GeV ≲ v ≲ 1016 GeV. singlets in 126 þ 126 leave the SUð5Þ subgroup unbroken The interpretation of the NANOGrav signal with and more Higgs fields are needed. The minimal Higgs 10 cosmic strings from a GUT phase transition indicates fields that break SOð Þ to GSM × M are then the existence of intermediate steps [50–54] in the breaking 45 þ 54 þ 126 þ 1264 and we will focus on this choice 10 → of SOð Þ GSM × M (with M the matter parity) throughout the paper. The two Higgs doublets Hu, Hd in because the SOð10Þ breaking also predicts monopoles the MSSM are chosen to be bidoublets in 10 for the that may overclose the universe. In a natural cosmologi- simplicity of analysis. cally safe scenario, monopoles are only produced during To summarize, we consider SUSY SOð10Þ models with symmetry breaking at high energy scales and get diluted the following set of heavy Higgs fields [57], away by inflation afterward. The remaining intermediate ¯ gauge group is finally broken to GSM at a lower scale S ¼ 54;A¼ 45; Σ ¼ 126; Σ ¼ 126: ð1Þ v ≳ 1014 GeV where cosmic strings emitting GWs are formed. The most general renormalizable superpotential of the In this paper, we study the consequences of such Higgs sector is intermediate scales in a supersymmetric SOð10Þ theory. The particles with masses below the unification scale λ mS 2 S 3 mA 2 λ 2 largely alter the renormalization group (RG) evolution of WH ¼ 2 TrS þ 3 TrS þ 2 TrA þ TrA S the gauge couplings from that in MSSM. Interestingly, it ΣΣ¯ η Σ2 η¯ Σ¯ 2 η ΣΣ¯ turns out that the unification of gauge couplings enforces þ mΣ þ S S þ S S þ A A þ; ð2Þ high intermediate scales that are needed by the cosmic 10 string interpretation of the NANOGrav result. Thus, the where we have omitted the terms with . We find minima of the Higgs potential by solving the F- and D-flat

2 conditions. We express the VEVs of GSM singlet fields The NANOGrav result is apparently in tension with the in terms of representations under the Pati-Salam previous results by other pulsar timing arrays EPTA [29] and ≡ 4 2 2 PPTA [30]. However, it is argued in [28] that this tension is the group G422 SUð ÞC × SUð ÞL × SUð ÞR to which they result of the improved treatment of the intrinsic pulsar red noise. belong: A recent solution invoking partially inflated cosmic string during inflation is given in [31]. 4 3Other interpretations of sources to generate GWs to explain The choice of 45 þ 54 þ 126 þ 126 is “minimal” in the the NANOGrav signal include phase transitions [36–40],pri- number of components. Another popular choice is 210 þ 126 þ mordial black hole formation [41–46] and dynamics of axionlike 126 [55,56] which leads to the “minimal” number of parameters particles [38,47]. in the general Lagrangian.

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TABLE I. The summary of patterns of VEVs at local minima of TABLE II. The mass spectrum of S, A, Σ, and Σ¯ Higgs fields in 10 ✓ 1 1 the potential that lead to a subgroup H of SOð Þ ( denotes a the model A. ð ; Þ0;1;2 denotes charges under the SM gauge ≡ 3 2 1 nonzero VEV). Stars in the first column indicate that the groups GSM SUð ÞC × SUð ÞL × Uð ÞY . symmetry breaking H → GSM × M is associated with the for- mation of cosmic strings. A circle in the first column indicates the States Mass scale formation of monopoles and cosmic strings at the same time, ¯ Σð10; 3; 1Þ, Σð10; 3; 1Þ MC which is not of our interest. Color triplets and sextets of Σð10; 1; 3Þ, MC Σ¯ 10 1 3 Hsabσ ð ; ; Þ Color triplets of Að15; 1; 1Þ MC SOð10Þ 1 1 1 1 Σ 10 1 3 ð ; Þ0, ð ; Þ1 from ð ; ; Þ, MR ∘SUð4Þ × SUð2Þ × SUð2Þ ✓ Σ¯ 10 1 3 ⋆ 3 2 2 1 ✓✓ ð ; ; Þ h i SUð Þ × SUð Þ × SUð Þ × Uð Þ M2 M2 A color octet and a singlet of ≡ R C ⋆SU 4 × SU 2 × U 1 ✓✓ M1 Max M ; M ð Þ ð Þ ð Þ Að15; 1; 1Þ C X SUð5Þ × M ✓✓✓ 2 ð1; 1Þ0, ð1; 1Þ 2 from Σð10; 1; 3Þ, M2 ≡ M =M G × M ✓✓✓✓ R X SM Σ¯ ð10; 1; 3Þ All the other components MX s ¼hSð1; 1;1Þi;a¼hAð15;1; 1Þi;b¼hAð1;1; 3Þi; Table II.5 Note that some states have masses different from σ ¼hΣð10;1;3Þi; σ¯ ¼hΣ¯ ð10; 1; 3Þi: ð3Þ the scales MX;C;R. The mass scale M2 is always smaller than M1 , while the order of M1 and M is not fixed in general. Patterns of VEVs which satisfy the minimum condition and ;R R When we perform multiple steps of RGE running, we lead to a subgroup H of SO 10 are summarized in Table I. ð Þ assume all particles at an energy scale Q share the same Note that the D-flatness sets σ σ¯ . j j¼j j mass Q. Generally, their masses depend on coupling Motivated by the result of NANOGrav, we focus on constants and the nonuniform spectrum generates a thresh- symmetry breaking patterns where cosmic strings form at old correction [58,59]. We discuss this point in the an intermediate scale M ∼ 1014−16 GeV. As indicated by R Appendix. stars in the first column of Table I,therearetwo The model B is obtained when jsj ≫ jbj ≫ jσj and possible choices of H whose breaking into G × M is SM a ∼ σ 2= s ≪ σ . However, it turns out that typically associated with the cosmic string formation without j j j j j j j j ≡ 3 2 the unification scale is lower in this model than that accompanying monopoles: G3221 SUð ÞC × SUð ÞL × 2 1 ≡ 4 2 1 in the model A. As a result, the current constraint from SUð ÞR × Uð ÞB−L and G421 SUð ÞC ×SUð ÞL ×Uð ÞR. When there is a hierarchy between VEVs of Eq. (3),the the proton decay rate is more severe and only a tiny region remains unconstrained. Thus, we will not consider this multistep breaking of SOð10Þ with at least one intermediate scale is achieved. The simplest possibility possibility and focus on the model A in the following is the breaking with one intermediate scale, i.e., discussion. 10 → → Finally, some comments on the fermion masses are in SOð Þ H GSM × M, where the breaking scale of H is identified as M . A detailed analysis of this order. It is well known that if the MSSM higgs doublets R 10 126 possibility is given in the Appendix, where we show Hu and Hd are embedded only in the (or ), the SM that it is difficult to construct a viable model that fermion masses at the GUT scale will be given by ml ¼ −3 survives collider constraints of the SUSY particle search. md (or ml ¼ md) for all three generations [60],which Thus, in the following discussion, we focus on models is in tension with the measured light fermion masses at with two intermediate scales, with the lower scale low scales. However, these relations of masses are altered identified as M ,andshowthatM ≳ 1014 GeV is by loop effects and higher-dimensional operators sup- R R – compatible with the unification of gauge couplings. pressed by the Planck scale [61 63]. In a renormalizable There are only two possible breaking patterns that setup, the MSSM Higgs doublets can be taken as linear 16 126 lead to surviving cosmic strings, SOð10Þ → G422 → H → combinations of doublets in or which renders the mass relation at the GUT scale arbitrary. However, it is GSM × M with H ¼ G3221 (model A) or H ¼ G421 (model B). impossible to mix the two doublets in the minimal The model A is obtained through the hierarchical choice choice of 45 þ 54 þ 10 þ 126 þ 126 for the Higgs sector of VEVs given by jsj ≫ jaj ≫ jσj, while jbj ∼ jσj2=jsj ≪ with a renormalizable superpotential [64]. One economi- jσj is ensured from the minimization condition of the cal solution is to introduce a 120 so that 10 mixes potential. We can define the unification scale MX, the Pati- → Salam breaking scale MC, and the G3221 GSM × M 5 ∼ ∼ Note that there can be mixing among the Higgs doublets in breaking scale MR through MX jsj, MC jaj, and 10, Σ, and Σ¯ , and two of them obtain light masses to be MSSM MR ∼ jσj, respectively. We summarize the mass spectrum ¯ Higgs doublets. The other Higgs doublets obtain masses of of all the components of S, A, Σ, and Σ Higgs fields in OðMXÞ, which are summarized in the last column of Table II.

035031-3 CHIGUSA, NAKAI, and ZHENG PHYS. REV. D 104, 035031 (2021) with 126 by the superpotential terms 45.10.120 and TABLE III. The coefficients of the RG equations of the gauge 45:126:120 [64].6 Since there are several approaches coupling constants for each energy range in the model A. Note θ ≡ Θ − θ ≡ Θ − Θ toward the SM fermion spectrum, we assume the that R ðQ MRÞ and 1 ðQ M1Þ with being the Heaviside step function. MSSM Higgs doublets lie mostly in 10 for the simplicity of the RGE analysis, and leave it open to specify the Energy range ðkÞ ðkÞ ðkÞ (k) b1 b2 b3 mechanism of realizing the measured fermion masses. MZ gaugino masses fixed at Oð1Þ TeV ute to the running from M2 to MR. At the G3221 breaking with other superpartners of SM particles at MS. In the 2 1 middle panel, we take account of typical sizes of GUT-scale scale MR, the gauge couplings of SUð ÞR and Uð ÞB−L, denoted as α2R and αB−L, respectively, are matched to that threshold corrections (see Appendix for the details). 7 of the SM hypercharge with The colored contours correspond to the choices MR ¼ 4 × 1014 GeV (red), 1015 GeV (green), and 4 × 1015 GeV 3 −1 2 −1 −1 (blue), while the solid and dotted lines correspond to α ðM Þþ α ðM Þ¼α ðM Þ; ð4Þ 5 2R R 5 B−L R 1 R tan β ¼ 2 and 50, respectively. The dependence on tan β comes from the two-loop contribution to RG equations α α and 2R and B−L run further up to the G422 breaking scale from Yukawa couplings. Note that in a concrete model, MC. The matching condition at MC is tan β is constrained by the fermion mass spectrum. Nevertheless, it has little impact on RG running and we α α α 4ðMCÞ¼ 3ðMCÞ¼ B−LðMCÞ; ð5Þ will use tan β ¼ 2 as a representative value. The lower gray region is excluded by either M MR. The coefficients bi are sum- 3;4;5 b4 ≈ b2 ¼ b2 . This effect is overwhelmed by uncertain- marized in Table III. In particular, bð Þ are written in a L R i ties we further discuss in the remaining of this paper. compact form within M2

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FIG. 1. Contour plot of MR as a function of MS and αU. Left panel: universal soft masses without threshold correction. Middle panel: universal soft masses with typical threshold corrections at the GUT-scale. Right panel: split SUSY spectrum with masses of gauginos 14 15 fixed at Oð1Þ TeV and without threshold corrections. The red, green, and blue lines correspond to MR ¼ 4 × 10 GeV, 10 GeV, 15 4 × 10 GeV, respectively, with tan β ¼ 2 (solid) and 50 (dotted). The lower gray region is excluded for either MC nucleon decays inside compact stars by GUT monopoles. weaker than that on the p → Kþν¯ mode, we focus on the Oð1Þ factor. As a whole, the above estimation of the proton latter. The proton lifetime is roughly given by [72] decay rate induced by the colored Higgses is expected to bear uncertainty of Oð1 ∼ 10Þ in each direction and can be 2 2 35 4 MS MHC worse for extremely small or large couplings in the τ → þν¯ ∼ 10 yrs × sin 2β : p K 105 GeV 1016 GeV superpotential. Another important contribution to proton decay comes ð7Þ from dimension-6 operators induced by the heavy gauge boson exchange [73,74]. Relevant gauge bosons are those In the following estimation, we will simply approximate the that transform under G as ð3; 2Þ5 6 and ð3; 2Þ−1 6. colored Higgs mass M ≈ M . SM = = HC X Compared with SUSY SUð5Þ models, the number of gauge The dimension-5 proton decay rate is model dependent bosons that contribute to the proton decay doubles, and the and Eq. (7) should be regarded as a rough estimate with proton decay width to the most important decay mode, large theoretical uncertainty. First, the size of the coupling p → π0eþ, increases. The proton lifetime is given by [75] between a colored Higgs and SM fermions is determined by the Yukawa coupling at the GUT scale, which is a source of 2 4 34 0.04 MX β τ →π0 þ ∼ 5 × 10 yrs × ; ð8Þ the tan dependence. However, values of Yukawa cou- p e α 1016 GeV plings highly depend on the structure of the Yukawa sector U 10 in terms of SOð Þ superfields, while we use typical values where we simply assumed that all the relevant dimension-6 of Yukawa couplings obtained by running the MSSM RG operators are mediated by a gauge boson with mass MX for equations in the estimation. Secondly, the intermediate a model-independent analysis. Note that the proton lifetime scales alter the running of Wilson coefficients from those in is highly sensitive to the gauge boson mass as can be seen 5 SUSY SUð Þ models [72] with which we perform our from the fourth-power dependence in Eq. (8), and thus the calculation. However, this effect is not significant due to the model dependent analysis of the GUT spectrum will be proximity between MR and MX, and is overwhelmed needed for a more precise estimation, which is however out by the uncertainty in the mass spectrum of sfermions. of the scope of this paper. In the evaluation, we have used Furthermore, Eq. (7) depends on the squared masses M2 Hc the RGE factor of Wilson coefficients valid for the MSSM 10 of the colored Higgses in . After setting the doublets running up to MX [76], which can be slightly modified due to the weak scale, MHc is a linear function of the vev to the difference of RGEs above MR. s ∼ OðMXÞ and the dimensionless coupling of the super- In Fig. 1, we show the current lower limit on the potential 10.10.54. If the dimensionless couplings are taken proton lifetime from the Super-Kamiokande experiment, 33 O 1 τ þ 5 9 10 to be ð Þ, the color triplet mass MHc can also vary by an p→K ν¯ > . × yrs [14], with black vertical lines.

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The lines correspond to tan β ¼ 1, 3, 5, 10, 30 and 50 from where vm is the relative velocity between the monopole and left to right. The region to the left of each line with smaller the Earth. The MACRO monopole search puts a constraint β −15 −2 −1 MS is excluded for each tan . Therefore, the left blue on the flux, Fm ≲ 1.8 × 10 cm s , which again sets β 1 region, whose right boundary corresponds to tan ¼ ,is the limit mm=TR ≳ 40. Assuming the critical temperature Tc β ≥ 1 → excluded for all tan . The upper gray region is the of the G3221 GSM × M phase transition is around the parameter space that is excluded by the current limit of symmetry breaking scale, TR ≳ Tc ∼ MR is required for the 34 τ 0 1 6 10 p→π eþ > . × yrs [13], while the dot-dashed line production of cosmic strings. We thus find a constraint on represents the prospect of Hyper Kamiokande [15] as an the hierarchy between the two intermediate scales, example of the future experiments. Note that the GUT scale MC=TR ≳ MC=MR ≳ 40α4. This constraint is shown in is generally large in the middle panel of Fig. 1. As a result, Fig. 1 with dashed lines; the regions below the dashed lines the constraint from p → π0eþ is absent in this panel. are constrained. The constraints above are trivial for GUT monopoles since their density is more exponentially suppressed. V. MONOPOLE DENSITY However, GUT monopoles catalyze baryon number viola- 8 In a series of phase transitions, SOð10Þ → G422 → tion processes [84,85] in compact stars and increase their → G3221 GSM × M, we assume that inflation occurs before luminosity. Various stringent bounds have been obtained in → the G3221 GSM × M phase transition and the reheating the literature [83]. The observation of old white dwarfs 0 −18 −21 −2 −1 temperature TR is above the critical temperature of this phase constrains Fm ≲ 10 ∼ 10 cm s [86,87]. The 0 transition so that cosmic strings that emit GW are populated measured x-ray fluxes of old nearby pulsars constrain Fm ≲ −22 −28 −2 −1 in the Universe. On the other hand, the breaking of G422 → 10 ∼ 10 cm s [88–92]. We note that the bounds G3221 generates intermediate scale monopoles with mass are subjected to large astrophysical uncertainties. In par- ≃ α mm MC= 4 [77,78]. To avoid the overclosure of the ticular, the quoted stronger or weaker end of the upper Universe, we require TR photon number density. Such monopole ρðlocalÞ ρ ∼ 105 ρðlocalÞ ≃ pole enhancement as DM = DM , with DM density is constrained in several ways. The dark matter 0 3 3 ρ ≃ cannot be solely made of monopoles because the local . GeV=cm the local dark matter halo density, DM 1 2 10−6 3 monopole density would be in severe contradiction with the . × GeV=cm the average dark matter density in the null result of monopole searches such as MACRO [81].We Universe, and Fm the flux if the GUT monopole were can then require Ω ≪ Ω , where Ω and Ω are the uniformly distributed throughout the Universe. The upper m M m M 0 ≲ 10−18 ∼ monopole and dark matter densities normalized by the bounds on the parameter space set by Fm 10−28 cm−2 s−1 are shown in Fig. 1 with orange bands critical density of the Universe. This sets mm=TR ≳ 40 for ∼1 GeV. Another equally strong constraint comes solely that rule out the regions below them. In most cases, the from the direct search of monopoles by MACRO. The bound from the GUT monopole is weaker than that from intermediate scale monopoles can be accelerated by the the intermediate scale monopole. The exception only ∼ 104 galacticpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi magnetic field [82,83] to a high velocity of order occurs when MS GeV with threshold effects 10−3 1017 GeV=m . Thus, they are not bounded to included for the RG evolution. m → galaxies and we assume them to be uniformly distributed If the G3221 GSM × M phase transition is of the strong in the Universe, with a flux near the Earth, first-order and experiences a supercooling phase, produced monopoles are diluted and the monopole density is sup- n =nγ v pressed. In this case, the constraint discussed above is F ≃3.7×10−17 cm−2 s−1 × m m ; m 3×10−27 300 kms−1 8 The intermediate scale monopoles from the breaking of G422 ð10Þ do not catalyze these processes.

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14 irrelevant. The abundance of cosmic strings produced at MR ¼ Oð10 Þ GeV, which is favored by the recent such a first order phase transition may be different from that numerical simulation of cosmic strings to explain the of a second-order transition. However, it hardly affects the NANOGrav data [32], with a moderate choice of MS. GW generation because it has been known that a network Because of the high cosmic string scale, the monopole of cosmic strings reaches a scaling solution [93]. abundance also places nontrivial constraints for the low αU region, even if its initial density is diluted by inflation. This VI. DISCUSSIONS complements the proton decay constraints at the high αU region. In conclusion, the GW observation gives us a way to For a given mass spectrum of supersymmetric particles, probe the supersymmetric grand unification. To extract the precise gauge coupling unification relates the intermediate intermediate scale M from the GW signal precisely, it is scale M of cosmic strings to the SOð10Þ breaking scale R R essential to reduce the uncertainty for the initial loop size α MX or the size of the unified gauge coupling αU. In Fig. 1, we compare the two extreme cases of the universal soft in the cosmic string spectrum. masses MS (left panel) and the split SUSY spectrum with gaugino masses fixed at Oð1Þ TeV while the other super- ACKNOWLEDGMENTS symmetric particles have a universal mass MS (right panel). S. C. is grateful to Ofri Telem for useful comments and Since we do not expect large modifications to these results discussions. Y. N. would like to thank Kohei Fujikura, when we take account of the detailed spectrum beyond the Motoo Suzuki and Tsutomu T. Yanagida for discussions. universal mass, these results demonstrate the spectrum Y. N. is grateful to Kavli IPMU for their hospitality during dependence of the gauge unification conditions. It is worth the COVID-19 pandemic. J. Z. would like to thank Goran noting that the GUT-scale threshold corrections can play Senjanovic for pedagogical conversations on SUSY important roles. As discussed in Appendix B, the size of the SOð10Þ during his visit to TDLI in January. S. C. is corrections highly depends on the precise mass spectrum of supported by JSPS KAKENHI grant. J. Z. is supported the GUT-scale particles. For a demonstration purpose, we in part by the National NSF of China under Grants pick up two reference points and compare results in the left No. 11675086 and No. 11835005. S. C. is supported by and middle panels of Fig. 1. the Director, Office of Science, Office of High Energy ≳ 1014 The high cosmic string scale MR GeV inferred Physics of the U.S. Department of Energy under the by the NANOGrav signal is a generic feature of the SUSY Contract No. DE-AC02-05CH1123. SOð10Þ model as we demonstrate in Fig. 1. The left panel of Fig. 1, which corresponds to the universal soft masses, APPENDIX A: ONE INTERMEDIATE SCALE shows that the current constraint on the proton lifetime and also the monopole overproduction constraint require super- The simplest possibility of the multistep SOð10Þ symmetric particles at a low-energy scale under the con- breaking is the case with one intermediate scale, 10 → → dition of the precise gauge coupling unification without the SOð Þ H GSM × M. We focus on models with GUT-scale threshold corrections. There is a tiny allowed H ¼ G3221 or G421 where cosmic strings are formed at parameter region with MS ¼ Oð10Þ TeV and MR ¼ the intermediate scale MR as shown in Table I. 15 Oð10 Þ GeV which is consistent with the NANOGrav The hierarchical VEVs, jsj ∼ jaj ∼ MX ≫ jσj∼ β ≲ 3 ≫ ∼ 2 data. However, this region with tan is in tension with MR jbj M2 ¼ MR=MX, lead to the breaking pattern 10 → → the observed SM Higgs mass [94]. As suggested in the of SOð Þ G3221 GSM × M. The mass spectrum of middle panel, the GUT-scale threshold corrections can this model is obtained by taking the limit of MC → MX in change this conclusion completely. The GUT scale MX Table II. As in the case of two intermediate scales, the 0 þ 2 1 tends to be higher, which renders the p → π e decay gauge couplings of SUð ÞR and Uð ÞB−L are matched to unobservable unless αU ¼ Oð1Þ. There is a vast parameter that of the SM hypercharge at MR through Eq. (4). All the region unconstrained by current and future proton decay gauge couplings of G3221 run further to the unification scale experiments. In this case, the GW observation together with MX and unify into a unique value αU. Consequently, we a better understanding of cosmic string formation serves as obtain the one-loop level relationship between the gauge α α the only means to narrow down the scale MR and thus the coupling constants i at MZ and U as SOð10Þ breaking scale M and the size of the unified gauge X 2π 2π coupling α through the requirement of gauge coupling ð1Þ MS ð2Þ M2 U ¼ þ b ln þ b ln unification. For the case of split SUSY without threshold α ðM Þ α i M i M i Z U Z S correction, the right panel of Fig. 1 also shows a large M M ≳ 10 ðaÞ R ðbÞ X allowed region with MS TeV, which can be consistent þ bi ln þ bi ln ; ðA1Þ with the SM Higgs mass while explaining the stochastic M2 MR

GW signal. The search for the proton decay at the Hyper- ð1Þ ð2Þ ðaÞ Kamiokande will explore a large fraction of the allowed where bi and bi are given in Table III, while bi ¼ 57 5 1 −3 ðbÞ 9 1 −3 parameter space, and the decay will be observed when ð = ; ; Þ and bi ¼ð ; ; Þ. We solve the three

035031-7 CHIGUSA, NAKAI, and ZHENG PHYS. REV. D 104, 035031 (2021) equations (A1) in terms of the three parameters MR, MX, Dynkin index of the representation R. Since these super- and αU, and obtain a set of solutions as a function of MS.It fields have the same SM quantum number and mix with is found that the correct hierarchy MZ

Eq. (A1), though in this case bðaÞ ¼ð97=5; 1; 2Þ and X i akðMðRÞÞ ðbÞ λ bR ln ; B2 b ¼ð81=5; 1; 0Þ. Again we found that the hierarchy i ¼ i Π ˜ ð Þ i R j≠NGQj MZ

( −9 0 − 3 4 λ − 5 η 2 ξ3 σ2 2 . 5 ð ln ln A þ ln SÞ; for =a > a =s; Δλ13 ≃ 3 ðB5Þ −9 0 a − 3 4 λ − 5 η 2 ξ3 σ2 2 . þ ln sσ2 5 ð ln ln A þ ln SÞ; for =a < a =s;

ξ3 ≡ η η¯ λ Δλ −7 5 Δλ −9 0 where we have defined S S S S. In the middle with 12 ¼ . and 13 ¼ . , which are typical values panel of Fig. 1, we define the gauge coupling at the obtained when all the dimensionless couplings are equal to GUT scale as unity.

α1ðMXÞ¼αU; ðB6Þ 9The symmetry breaking scale is defined as the mass of gauge αU αiðMXÞ¼αU 1 þ Δλ1i ði ¼ 2; 3Þ; ðB7Þ bosons so Nambu-Goldstone bosons do not contribute to the 2π threshold correction.

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