Pacific 2019 @ Moorea 2019 Sep 3 (Aug 31- Sep 6) Hierarchy Problem: Implication to Cosmology (Supercooled universe) and Implication from Superstrings (Revolving D-branes)
Satoshi Iso (KEK & Sokendai)
1 Hierarchy problem = Dynamics of EWSB and its Stability
How EWSB occurs dynamically ? Who ordered the Higgs potential ? Why Higgs VEV
Cosmology early universe Particle physics DM, baryogenesis (LHC, ILC, …) Hierarchy TeV or beyond ? Problem Superstring unification Geometry of SM String threshold corrections
2 Today’s talk
More about Hierarchy problem and classically conformal model
[2] Implication to cosmology (the early universe) : Supercooled universe
[3] Implication from superstrings : String threshold corrections
3 Rough picture of the Higgs potential: (at least) 3 important points
Higgs potential
3 Energy scale 1 2 1 Higgs VEV v= 246 GeV à Particle physics (present universe) 2 UV scale e.g. MPlanck à Quantum Gravity, String 3 h=0 (origin) à Cosmology (early universe)
All regions (1, 2, 3) are related by RG and theoretical biases.
4 2 important lessons from LHC (1) mass = 125 GeV "EW" physics may be directly related to Planck scale physics without intermediate scales in between. (2) No deviations from SM / no TeV SUSY?
New paradigm à classical conformality (CC)
EWSB occurs, NOT by (classical) Higgs potential V(h) but by radiative corrections to it. Coleman-Weinberg mechanism
5 For the classical conformal idea to the hierarchy problem two conditions are necessary
1. No intermediate scales = EW directly connected to Planck. 2. String threshold corrections are controllable. (or Planck scale does not destabilize Higgs potential.)
The first condition is suggested by Higgs mass 125 GeV.
The second condition needs something more. I will show one example of non-SUSY vacua in superstring that satisfies the second condition.
6 An example of a bottom-up model of CC An additional sector to the SM is inevitably needed. Within SM, CW mechanism predicted Higgs mass lighter than 10 GeV.
Okada, Orikasa, SI (09) A phenomenologically viable minimal extension will be also Lindner et.al. (10) (B-L)-extension of SM
IR physics B-L sector @ TeV UV physics Standard U(1)B-L gauge Planck scale Model + SM singlet scalar V(h)=0 Right-handed ν No intermediate scales
(1) B-L breaking by CW mechanism à triggers EWSB at 100 GeV Everything occurs radiatively. (2) Minimal extension of SM & Phenomenologically viable All new particles are predicted to be around EW - TeV 7 [1] Implication to Cosmology
Thermal history of the universe Serpico, Shimada, SI (17)
“Standard” Crossover Crossover
T 100 GeV 100 MeV MeV ⇠ ⇠ ⇠ Inflation EW QCD phase transition phase transition BBN
Reheating extremely supercooled (strong first order PT) χSB induces EWSB
→ cosmological consequences: GW, DM, PBH ... m2 =2λ h 2 m 1/3 yt tt¯ vQCD = ⟨ ⟩ ΛQCD !√2λh " ∼ g>0.2 g<0.2 After primordial inflation, hot universe starts at T>> TC . V (φ) (g2T 2 λ v2 /2)φ2 ∼ QCD − | mix| QCD Potential of the B-L scalar High- T β > 03m4 > 2Tr(m4 )+Dm4 D 41 m afewhundredGeV φ Z′ N h Z′ ∼ ∼ Ti mZ dλmix 1 mRHν,mφ mZ afewhundredGeV1M N =ln QCD ln QCD′ 2 2 ′ ⊙ T T = 6gB Lgmix + λmix( ) ≪ ∼ N ∼ N dt 16π2 − ··· T = T m ! " c Z′ 2 2 nd ∼ λmix gB Lgmix 2 inflation starts ∼− − V0 mZ′ ≃ valley 9 1 1 m2 =2λ h 2 m 1/3 yt tt¯ vQCD = ⟨ ⟩ ΛQCD !√2λh " ∼ g>0.2 g<0.2 V (φ) (g2T 2 λ v2 /2)φ2 Potential for B-L scalar field ∼ QCD − | mix| QCD High- T (h=0, φ=0) β > 03m4 > 2Tr(m4 )+Dm4 D 41 m afewhundredGeV φ Z′ N h Z′ ∼ ∼ T m Trapped in i Z′ dλmix 1 2 2 mRHν,mφ mZ′ afewhundredGeV1M N =ln QCD ln QCD ⊙ TN TN = 2symmetric6gB Lgmix + λmix( ) ≪ ∼ ∼ dt 16π − ··· T = T m ! " c Z′ phase for2 2 nd ∼ λmix gB Lgmix 2 inflation starts a ∼−long time− V0 mZ′ 2 2 ≃ m =2λ h mφ 10 1 1 1 m2 =2λ h 2 m 1/3 yt tt¯ vQCD = ⟨ ⟩ ΛQCD !√2λh " ∼ Supercooling of (B-L)+EW à PT much belowg>0.2 Tg V (φ) (g2T 2 λ v2 /2)φ2 ∼ QCD − | mix| QCD High- T (h=0, φ=0) β > 03m4 > 2Tr(m4 )+Dm4 D 41 m afewhundredGeV φ Z′ N h Z′ ∼ ∼ T m Trapped in i Z′ dλmix 1 2 2 mRHν,mφ mZ′ afewhundredGeV1M N =ln QCD ln QCD ⊙ TN TN = 2symmetric6gB Lgmix +tunnelλmix( ) ≪ ∼ ∼ dt 16π − ··· T = T m ! " c Z′ phase for2 2 nd ∼ λmix gB Lgmix 2 inflation starts a ∼−long time− V0 mZ′ 2 2 ≃ m =2λ h mφ 11 1 1 1 Percolation of true vacuum Guth Weinberg (82) Bubbles of true vacuum are created by tunneling Tp : percolation temperature Universe is occupied with true vacuum bubbles 12 critical bubble action TP = percolation temperature S/T ~ 1/g3 can be calculated by tunneling rate. Serpico, Shimada, SI (17) B-L gauge coupling If g < 0.2, universe is never occupied with true vacuum until T decreases down to 100 MeV or much lower. Note that de Sitter fluctuation T = / 2 ⇡ is negligible at 100 MeV. ⇠ GH H 1/4 V m 13 0 ⇠ Z0 But when temperaturem2 =2decreasesλ h 2tom 100 In the present case, since h=0 2 2 2 2 V (φPisarski) Wilczek(g T(1983)QCD λmix vQCD/2)φ all Nf =6 quarks are massless ! ∼ − | | à 1st order phase transition for N 3 f β > 03m4 > 2Tr(m4 )+Dm4 D 41 m afewhundredGeV φ Z′ N h Z′ 14 ∼ ∼ T m m ,m m afewhundredGeV1M N =ln i ln Z′ RHν φ Z′ T QCD T QCD ≪ ∼ ⊙ N ∼ N T = T m c ∼ Z′ 1 m2 =2λ h 2 m V (φ) (g2T 2 λ v2 /2)φ2 ∼ QCD − | mix| QCD β > 03QCDm4 χSB> 2Tr(triggersm4 )+ BDm-L and4 D EWSB41 m afewhundredGeV φ Z′ N h Z′ ∼ ∼ T m m ,m m afewhundredGeV1M N =ln i ln Z′ RHν φ Z′ T QCD T QCD ≪ ∼ ⊙ N ∼ N T = T m c ∼ Z′ 15 1 1 1 Cosmological consequences [ QCD + (B-L) + EW ] strong 1st order phase transition expected Bubble collsions à Sizable Stochastic Gravitational Waves from 1st order PT 2 h ⌦GW,0 100-10 C3 0.6 ⇣ 0.3 C2 11-11 ⇡ 10 10 ⇣ 0.3 / = 10 C1 ⇡ H eLISA sensitivities 0.15 from C.Caprini et al. (2016) 12 10 10-12 / = 100 H 13 10-13 ⇣ 0.3 10 ⇡ / = 1000 Serpico, Shimada, SI ('17) H -5 -4 Jinno, Takimoto ('17) 10 10 4 0.001 3 0.010 2 f0/Hz0.100 10 10 10 Hashino et.al. ('18) ...... Many other implications dilution factor 106 dilutes relics from high energy (high temperature) WIMP DM, baryon asymmetry à supercool DM Hambye Strumia Teresi(18) nd low reheat temperature after 2 mini-inflation ( TR < 100 GeV) Baryon asymmetry at low T: e.g. Cold EWBG Konstandin, Servant (11) .... Enhanced scalar fluctuations PBH by 1st order QCD PT or Ultra-compact mini halo Jedamzik (97), Ricotti et(09)16 m2 =2λ h 2 m 1/3 yt tt¯ vQCD = ⟨ ⟩ ΛQCD !√2λh " ∼ Summaryg>0.2 g< of0. 2 [1] Serpico, Shimada, SI (17) In classically conformal models motivated by naturalness, V (φ)=(g2T 2 λ v2 /2)φ2 the early universe is drasticallyQCD different− | mix | QCD extremeβ > 03 mSUPERCOOLING4 > 2Tr(m4 )+andDm the4 Dsecond41 inflationm afewhundredGeV with N~10 φ Z′ N h Z′ ∼ ∼ T m m ,m m afewhundredGeV1M N =ln i ln Z′ RHν φ Z′ T QCD T QCD ≪ ∼ ⊙ N ∼ N 100 MeV χSB (B-L)+EWSB Interesting cosmological consequences Stochastic GW, PBH, Cold EWBH, axion abundance , supercool DM Supercooling is also expected in other models, e.g. Randall-Sundrum models, Harling, Servant (18) 17 1 [2] Implication from Superstrings N. kitazawa, H.Ohta, T.Suyama, SI to appear also N. Kitazawa, SI PTEP (15) H. Ohta, T. Suyama, SI arXiv:1812.11505 N. Kitazawa, SI arXiv:1812.08912 18 In the top-down stringy approach, we want to answer the following questions about Higgs: (1) How can we embed a bottom-up model in string theory? “stringy realization of the SM and the Higgs field” (2) How can we control threshold corrections of Planck scale physics ? In string theory, there are many superheavy particle which may produce large radiative corrections to Higgs potential 2 2 (MUV) H MUV : string scale or SUSY breaking scale = hierarchy problem in string theory 2 2 In what follows, I will show one example of (MUV) H =0 in string th. 19 What is the Higgs potential? In a field theoretic approach, we usually construct a Higgs potential first. "Consider a Higgs potential first !" Coleman-Weinberg type Then obtain a solution as a minimum of the potential. one solution to one Higgs potential à We are faced with the naturalness problem 20 Question: Can we obtain a solution of Analogy with orbital motion of planet There are many stationary solutions. Potential is calculated for each solution. Each orbits corresponds to a different angular momentum. “Dynamical-tuning of Higgs potential” 21 How can we embed Higgs (and SM) in superstring theory ? In string theory, dynamics of field theory is transformed into a problem of geometry. D-branes, moduli, compactification, … Hierarchy problem also becomes geometrical. 22 V = µ2 H 2 + λ( H 2)2 − | | | | 1 d4k δV = Str log(k2 + M(φ2)) 2 (2π)4 ! Λ2 M(φ2)4 M 2 1 = StrM(φ2)+Str ln 32π2 64π2 Λ2 − 2 " # 2 StrM(φ2)=0?Λ = M exp( 8π ) TC TC − bg2 r lstring ω mstring r≪ ω 1≪ lstr ∼ mstr ≪ V r 2 ω J = Mr2ω = (1) glp l m str " str # " str # v = rω 1 ≪ 2 2 2 2 2 1 J 1 2 mstr r ω j lstr 2 = mv = = (2) V 2Mr 2 2g lstr mstr 2g r One possibility " # " # " # mstr V M = p (3) D-Brane world scenario g lstr Λ2 M(φ)4 M 2 1 [1] (3+1)-dimensional space-time= is embeddedStrM(φ)2 in+S d=9+1.tr ln (4) 32π2 64π2 Λ2 − 2 D3 brane " # [2] Higgs (scalar) rfieldl smaytr be a geometrical "moduli" field ≫ 2 distance between D-branes = Higgs vevStrM(φ) =0 (5) Mv4 J 2 − + (6) r7 p Mr2 N1 − D-branes v4 M d d = (7) U(N1 + N2) à U(N1) U(N2test) M 2 " # s N2 How can we obtain a small value of d << lstring ? = Hierarchy problem in string theory For phenomenological viability, we need more complicated models,23 In the following, we consider the simplest set-up as a1 toy model. D-brane universe Suppose that many D-branes were moving randomly in the early universe. e.g. D-brane inflation Open strings Due to nonadiabatic particle productions (=parametric resonance), D-branes lose their kinetic energy. “Beauty is attractive” by Kofman et.al. 24 What is the fate of D-branes ? Enomoto et.a. (2014) In the bosonic string, they form a bound state (if closed string emission is neglected. ) 25 In superstring theory, if D3 branes are at rest in parallel, it is BPS. D3 in d=10 velocity v distance r v4 7 p r − No interaction Attractive potential is generated (BPS) by one-loop string amplitude Angular frequency w Can they form a bound state ? (cf. D0s: threshold bound state ) Rotational motion breaks SUSY à SUSY breaking scale = w 26 1 In this paper, we propose an approximate method to calculate the threshold corrections of open string massive modes. The method is to combine the SYM and the supergravity calculations with an appropriate cutoff. The method was first suggested in [DKPS], and uses a partial modular transformation (open-closed string duality). The partial modular transformation avoids a double counting. We then applied the method for calculating potential between revolving D3-branes and show that the expected threshold corrections ω2r2 vanish. The first nontrivial threshold ∼ 4 2 2 corrections from massive open string modes are given by (ω /mstr)r , which are 2 ∼ suppressed by a factor of (ω/mstr) . The paper is organized as follows. In sec 2, we first introduce an approximate method to calculate string threshold corrections from massive open string modes in D-brane models. In sec 3, we apply the method and calculate effective potential of the revolving Dp-branes. The contributions from massless open string modes are calculated in the SYM theory with a stationary revolving background. In sec 4, we calculate the contributions from massive open string modes using the supergravity theory with an appropriate Schwinger parameter cutoff. In sec 5, we explicitly eval- uate the effective potential V at short distance r m by expanding the formulae ≪ str derived in previous sections, sec 3 and sec 4, and then discuss a possibility of a bound state. 2 String Threshold Corrections in D-brane Models We are interested in calculating interaction potential between D-branes, which are relatively moving in a target space-time. At weak string coupling, we can obtain the effective potential by calculating one-loop partition function of an open string stretched between the D-branes. For some simple cases, we can quantize the stretched open strings exactly and determine a closed form of the one-loop effective potential. But in many other cases where D-branes are accelerating, it is not possible to write the effective potential in a closed form since open strings have complicated bound- ary conditions. For example, when two D-branes are revolving like a binary star, open string spectrum can be solved only perturbatively with respect to the relative Interaction potential between revolving D3-branes velocity [IOS]. Thus, in order to calculate the potential between these D-branes, it is necessary to develop more clever methods. In this section, we propose a method to obtain interaction potential betweenSuppose generally that moving two parallel D-branes D3s including thresh- are rotating in a transverse plane old corrections of massive open string modes. The method was first indicated in a seminar paper [DKPS].Calculation of the potential in string theory Schematically,= one the-loop effective open string potential amplitudeV ( withr) is rotating given boundary as condition 2 ∞ dt r t V (r)= e− 2πα′ Z(t), (2.1) − t !0 where r is the distanceZ(t): partition between function the D-branes.of open stringZ(t) is the partition function of the stretched open stringit is with generally the difficult modulus to quantize (Schwinger the open paramter)strings t, where the factor 2 r t because of complicated boundary conditions. e− 2πα′ due to the string tension is extracted. In many known examples, the r- 2 r t 27 dependence only appears through e− 2πα′ , and Z(t)isr-independent. The method 3 In field theories, the string calculation is interpreted as a sum of one-loop amplitudes of infinitelyv4 many fields 7 p massless (SYM) + massive fieldsvr4 − 4 v 7 p F 1 ∞ r − ∞ ( 1) r7 p F 2 2 2 2 4 2 2 V − ( 1) dN tr log(p + mN )= dN − 2 Λ mN + mN log mN /Λ ∼ 2 N=0 − N=0 64π ! ! F " # 1 ∞ F F 2 2 ∞ ( 1) 2 2 4 2 2 1 ∞ F 2 2 ∞ ( 1)2 2 2 2 42 2 2 2 2 2 V ( 1) trm log(N =pf(m+strm,rN,)=ω ) (Nm−str) + Λ(rm,Nω +) mN log mN /Λ V ( 1) dN tr log(p + mN )= dN − 2 Λ mN + mN log mN /Λ 2 ∼ 2 − ∼ 2 N=1 − 64π N∼=1 64π O N!=0 !N!=0 " !L2 # " # ω = 2 2 2 2 2 2 2 2 2 2 2T3r 2 2 2 mN = f(mstr,r , ω ) (Nmmstr) =+ f((mr , ω,r) , ω ) (Nm ) + (r , ω ) ∼ N O str ∼ str O L2 ω = 2 Massless2T states3r (SYM) may dominantly contribute to V(r). supersymmetry at w=0, V(r) ~ !"#" Infinitely many massive states can also contribute to V(r) = ∑ !"#" How to calculate the stringy threshold corrections ? = hierarchy problem 28 1 1 1 N.Kitazawa, H. Ohta, A new method to calculate the stringy T. Suyama, SI (09) threshold corrections in D-brane models Basic idea = Partial Modular Transformation (open-closed string duality) one-loop open string amplitude = closed string exchange à The stringy calculation is reduced to a partial summation of SYM +SUGRA (no double counting) The potential can be calculated with Less than 0.1% accuracy. 29 by a sum of contributions of supergravity fields: V˜ = 2κ2 dp+1ζ dp+1ζ˜∆(X X˜) F (X, X˜)+F (X, X˜)+F (X, X˜) , c − 10 − Φ g C ! ! " (4.12)# where p 3 2 F (X, X˜)= − T 2 detη ˆ (X) detη ˆ (X˜), (4.13) Φ 4 p − αβ − γδ $ % & & F (X, X˜)=T 2 detη ˆ (X) detη ˆ (X˜) g p − αβ − γδ & (p + 1)2 1& + ηˆαβ(X)(∂ X ∂ X˜)ˆηδγ(X˜)(∂ X˜ ∂ X) , × − 16 2 β · δ γ · α $ % potential in the Lorentzian signature up to (ω4); (4.14) O ˜ 2 ˜ 4 2 FC (X, X)=T4 p det(∂αX ∂βX). (4.15) Λ 4r 4r2/Λ2 16r 2 2 · V˜ (r)= 1 e− + E (4r /Λ ) o,B −4π2 − Λ2 ˜ Λ˜4 1 ˜ !" FΦ#(X, X), Fg(X, X), FC (X, X)$ are contributions from dilaton, graviton, and RR- 2 fields respectively.2 4 The details of the calculation are given in Appendix A. 2 r 4r2/Λ2 r 4r 2 2 ω e− + E (4r /Λ ) − π2 − π2 π2Λ2 1 ! " # $ 2 4 4 6 4 1 4.42r SUGRA10r potential4r2/Λ2 in Revolving6r 40r Dp-branes2 2 ω + + e− + E (4r /Λ ) − 24π2 3π2Λ2 3π2Λ4 − π2Λ4 3π2Λ6 1 !" # " # $ + (ω6), In the following, we apply the result (4.12) for a general(5.3) Dp-brane motion to the O revolving Dp-branes. The embedding functions Xµ and X˜ µ for the revolving Dp- 4 2 4 Λ 4r branes4r are2/Λ2 given16r by 2 2 V˜ (r)= 1 e− + E (4r /Λ ) o,F 4π2 − Λ2 Λ4 1 !" # Xα = ζα,X$8 = r cos ωζ0,X9 = r sin ωζ0, 2 2 (4.16) 2 r 2 2 4 ˜ α1 ˜α r ˜ 8 4r2/Λ2 ˜0 6 ˜ 9 ˜0 ω E (4r /Λ ) ω X = ζ , X =e− r cos ω+ζ , (ω )X. = (5.4)r sin ωζ . − π2 1 − 48π2 − 12π2Λ2 − O − ! $ !" # $ Inserting these functions into (4.12) and performing most of the integrations, we potential in the Lorentzian signature up to (ω4); Note that we have set 2παobtain′ = 1. The E1(x) isO the exponential integral defined in (B.9), whose small x behavior4 is2 given by 4 Λ 4r 4r2/Λ2 16r 2 2 4 V˜ (r)= 1 e− + E (4r /Λ ) 10 p v ∞ 10 p o,B 2 2 ˜ 4 2 1 2 2− 2− −4π − Λ Vc(r)=Λ κ10Tp Vp+1(4π)− 2 ds s− !" # − 2 $ 1 v ˜ 2 x 3 Λ− 2 2 4 − ! E1r(x)=2 γ2 logr x +4xr + (x ). (5.5) 2 4r −/Λ − − 4 O2 2 1 2 2 2 ω 2 e− 2 + 2 2 E1d(4ζrexp/Λ ) ζ +2r (1 + cos ωζ) (1 + cos ωζ) , (4.17) − π − π π Λ× −4s Both of the bosonic and! fermionic contributions" #! have quartic' $ and quadratic diver- * 2 4 4 ( 6 ) 4 1 2r 10r 4r2/Λ2 6r 40r 2 2 gences but they are completelyω where+ cancelledΛ˜ is a+ cut-o as expected.ff. e− The sum is+ given byE (4r /Λ ) − 24π2 3π2Λ2 3π2Λ4 − π2Λ4 3π2Λ6 1 !" # " # $ ˜ ˜ 2 2 Following2 the recipe in sec 2.4 , we can fix the cut-off Λ so that Vc(r) gives ω r 4r26/Λ2 4r 2 2 ˜ V˜o(r)= +e− (ω ),the correctE threshold1(4r /Λ ) corrections to Vo(r). The suppression(5.3) factor due to the string − π2 O − Λ2 r2/s ! 4 tension"2 in# the above4 integrand$ is given by e− .Thenthecut-off is given by s =1if Λ 4r 2 2 16r 2 V˜ (r)= 1 s is rescaled2 e 4r /Λ such4+ thatE this(4r2 factor/Λ2) becomes4 e 6(2r) /(2πα′s). This amounts to choosing o,F 12 7r2 − 10r 4 12 2 6r 40r− 4 4π −˜ Λ2 Λ 4r /Λ 2 2 ωEffective2!" potential+ Λ− 2=#2 παV(r)+′/ at2.2 fixed4 e w− : r =moduli$ 2 field4 + = Higgs2 6 ?E1(4r /Λ ) − 16π 2 12π Λ 3π Λ −2 π Λ 3π Λ !" r Several comments# 1 are inr order." First,2 2 let us# investigate$ the large r behavior of ω2 E (4r2/Λ2) ω4 e 4r /Λ + (ω6). (5.4) 6 20.2 1 0.4 0.6 0.8 2 1.0 2 2 − ˜ + (ω )-.−7 π the potential− with v48=π r−ω 12fixedπ LargeΛ as a r smallbehavior value.O is simple: In the limit (5.6) Λ , we obtain O -2.×10 ! $ !" # $ →∞ -7 4 Note that-4.× we10 have set 2πα′ = 1. The E1(x) is the exponential integral7 p defined inv The form of the potential is valid as far as the˜ condition 2ω The contributions from the supergravity in (4.17) can be also obtained in the ω expansion as JIGEN GA HEN. LAMBDA HA MASS NI SHITE HOSHII 4 ω 2 2 Λ˜ 2r2 6 V˜ (r)= 1 1+Λ˜ r e− + (ω ). (5.7) c −16π2 − O % & ' ( 20 At large r, it is approximated by V˜ (r)= ω4/16π2 = v4/16π2r4 and repro- c − − duces the Newton-like potential for D3 branes in D=10. At small r, it becomes 4 2 2 2 2 ω r Λ˜ /16π , which is suppressed by a factor (ω/mstr) compared to a naive ex- pected behavior ω2r2. The absence of large threshold corrections to the moduli mass has an interesting implication to the hierarchy problem of the Higgs potential. In the region r<ω, we can perform a different approximation of the integral for V˜o(r) to estimate the shape of the potential. Details of the calculations are given in the appendix C, but the leading order behavior is the same as the above ω expansion and given by ω2r2 V˜ (r) . (5.8) o ∼− π2 Thus as far as the leading behavior is concerned, (5.6) seems to give a good approx- imation at small r. Now we plot the total potential, V˜ = V˜o + V˜c, including both of the contributions from the SYM and theFurther threshold comments corrections and from future SUGRA directions in in Figure ( ). PLOT ha w-exp denaku integral sonomama yattahouga yoi, soreto w-exp kurabetemo yoiga (1) String threshold corrections # # 5.2 Can the revolving D3-branes∑ " $ = have0 !! a boundω : SUSY state? breaking scale Based on these expressions, we canString give threshold a rough argumentcorrections for (including whether there gravity) could vanish v4 exist a bound state of D3-branesopposed in the potential to a naïveV˜ (r ).field In thetheory argument expectation of a bound ! 7 p state, we assume that the angularHow momentum universal is is it? conserved. Thenr − we take into account the effect of the centrifugalLarge potential SUSY for in thethe D3-branes.bulk and D This-brane implies breaks that SUSY we need to analyze the behavior of the potential with fixing the angular momentumF 1 ∞ ∞ ( 1) L of the D3-branes, insteadV of ω. Therefore( 1) theF tr potential log(p2 we+ needm2 )= to study is− given Λ2m2 + m4 log m2 /Λ2 (2) ∼A possibility2 − of a bound state N 64π2 N N N by N=1 N=1 " # ! L2 ! U(r):= + V˜ (r) L : angular momentum(5.9) 4T r2 2 2 2 2 2 2 3mN = f(mstr,rfor, ω unit) volume(Nmstr of) D3+-brane(r , ω ) with ω replaced with L2/2T r2. The relative distance and reduced mass∼ for a unit O 3 Centrifugal potential L2 volume is given by 2r and T3. Since we are based on the one-loop string calculations, ω = 2 the string coupling constant should be smaller than 1. In such a situation,2T3r the potential U(r) behaves like in Figure ( ),Not and yet there conclusive is no minimum because the centrifugal potential is more dominant than the induced attractive force by one-loop 31 calculations. It excludes a possibility of forming a bound state for revolving two D3 branes. The situation is modified if we consider a stack of N D-branes revolving around each other. Suppose that each of the revolving D3-branes are replaced with N D3- branes. Then, V˜ (r)ismultipliesbyN 2 since there are N 2 open strings stretched between the two sets of D3-branes. On the other hand, the centrifugal potential is multiplied simply by N. Therefore, the potential U(r) will be modified as 2 NL 2 ˜ UN (r):= 2 + N V (r). (5.10) 4T3r For a sufficiently large N (e.g., N is larger than 5), the behavior of UN (r) changes to the figure drawn in Figure ( ), which is qualitatively different from U(r). As long 21 1 V = µ2 H 2 + λ( H 2)2 − | | | | 1 d4k δV = Str log(k2 + M(φ2)) 2 (2π)4 ! Λ2 M(φ2)4 M 2 1 = StrM(φ2)+Str ln 32π2 64π2 Λ2 − 2 " # 2 StrM(φ2)=0?Λ = M exp( 8π ) TC TC − bg2 r lstring ω mstring r≪ ω 1≪ lstr ∼ mstr ≪ V r 2 ω J = Mr2ω = (1) glp l m str " str # " str # v = rω 1 ≪ 1 J 2 1 m r 2 ω 2 j2 l 2 = mv2 = str = str (2) V 2Mr2 2 2g l m 2g r " str # " str # " # mstr V M = p (3) g lstr Λ2 M(φ)4 M 2 1 = StrM(φ)2 +Str ln (4) 32π2 64π2 Λ2 − 2 " # r lstr ≫ StrM(φ)2 =0 (5) (3) Experimental test ofMv the4 “stationary”J 2 scenario of Higgs − 7 p + 2 (6) r − Mr Lorentzv violation4 inM the Higgs sector d = d l ω m (7) (Coriolistest force forM Higgs2 ≪fieldstring since ≪it is geometrical.)string " # s 4ω2 ω4 ω2 = M 2 +(1+ 0 )p2 +16 0 p4 + N. Kitazawa,(8) SI ('18) M 2 M 6 ··· ω0 < 0.1 GeV Other SM particles living on the brane do not feel Coriolis force just like we do not feel rotation of the earth. 1 (4) More pheomenologically viable models 32 Conclusion QCD-induced EW PT Higgs physics early universe String theory @ LHC + ILC + ... space-time physics Lorentz violation in Higgs naturalness Dark matter moduli = geometry stability Baryogenesis SUSY breaking Yukawa couplings Inflation, PBH, ... Dark energy Hierarchy problem is a key idea to go beyond SM in particle physics, cosmology, string theory Implication to Cosmology: Supercooling of EWPT, Different scenario of DM, B, L genesis Implication from Superstring: stringy threshold corrections Mass terms of Higgs may not be generated 33 34