Realizing the relaxion from multiple axions and its UV completion with high scale supersymmetry
CCTPU-15-16
Realizing the relaxion from multiple axionsand its UV completion with high scale supersymmetry
Kiwoon Choi ∗ and Sang Hui Im † Center for Theoretical Physics of the UniverseInstitute for Basic Science (IBS), Daejeon 34051, Korea
Abstract
We discuss a scheme to implement the relaxion solution to the hierarchy problem with multipleaxions, and present a UV-completed model realizing the scheme. All of the N axions in ourmodel are periodic with a similar decay constant f well below the Planck scale. In the limit N (cid:29)
1, the relaxion φ corresponds to an exponentially long multi-helical flat direction whichis shaped by a series of mass mixing between nearby axions in the compact field space of N axions. With the length of flat direction given by ∆ φ = 2 πf eff ∼ e ξN f for ξ = O (1), both thescalar potential driving the evolution of φ during the inflationary epoch and the φ -dependentHiggs boson mass vary with an exponentially large periodicity of O ( f eff ), while the back reactionpotential stabilizing the relaxion has a periodicity of O ( f ). A natural UV completion of ourscheme can be found in high scale or (mini) split supersymmetry (SUSY) scenario with the axionscales generated by SUSY breaking as f ∼ √ m SUSY M ∗ , where the soft SUSY breaking scalarmass m SUSY can be well above the weak scale, and the fundamental scale M ∗ can be identifiedas the Planck scale or the GUT scale. ∗ email: [email protected] † email: [email protected] a r X i v : . [ h e p - ph ] J a n . INTRODUCTION Recently a new approach to address the hierarchy problem has been proposed in [1].The scheme introduces a scalar degree of freedom, the relaxion φ , making the Higgs bosonmass a dynamical field depending on φ . During the inflationary epoch, the Higgs bosonmass-square µ h ( φ ) is scanned by the rolling φ from a large positive initial value to zero.Right after the relaxion crosses the point µ h ( φ ) = 0, so that µ h ( φ ) becomes negative,a nonzero Higgs vacuum expectation value (VEV) is developed and a Higgs-dependentback reaction potential begins to operate to stabilize the relaxion . One can then arrangethe model parameters in a technically natural way to result in the relaxion stabilized ata point where the corresponding Higgs VEV is much smaller than the initial Higgs bosonmass.An intriguing feature of the relaxion mechanism is that the relaxion potential involvestwo very different scales. One is the period of the back reaction potential, and the otheris the excursion range of the relaxion necessary to scan µ h ( φ ) from a large initial value tozero. To see this, let us consider the relaxion potential given by V ( φ, h ) = V ( φ ) + µ h ( φ ) | h | + V br ( φ, h ) (1)where V is the potential driving the rolling of φ during the inflationary epoch and V br isthe periodic back reaction potential stabilizing φ right after it crosses µ h ( φ ) = 0. In fact,the key feature of the mechanism can be read off from the following form of potential: V = (cid:15) f φ + ...., µ h = M h + (cid:15) h f φ + ..., V br = Λ ( h ) cos (cid:18) φf (cid:19) , (2)where M h denotes the initial Higgs boson mass, (cid:15) and (cid:15) h are small dimensionless pa-rameters describing the explicit breaking of the relaxion shift symmetry in V and µ h ,respectively, and finally f is the relaxion decay constant in the back reaction potential.In non-supersymmetric theory, the Higgs mass parameter M h is naturally of the order of A mechanism to cosmologically relax the Higgs boson mass down to a small value through a nucleationof domain wall bubbles has been discussed in [2]. M h is much largerthan the weak scale: M h (cid:29) v ≡ (cid:104) h (cid:105) = 174 GeV , (3)which might be explained by the relaxion mechanism.Let us now list the conditions for the relaxion mechanism to work. First of all, in orderfor the rolling relaxion to cross µ h ( φ ) = 0 without a fine tuning of the initial condition, itshould experience a field excursion ∆ φf (cid:38) M h (cid:15) h f . (4)In order for the scalar potential to be technically natural under radiative corrections, thesymmetry breaking parameters (cid:15) and (cid:15) h should obey (cid:15) (cid:38) (cid:15) h M h f . (5)On the other hand, from the stability condition ∂ φ V = 0, one finds (cid:15) ∼ Λ f , (6)and therefore ∆ φf (cid:38) M h Λ . (7)As for the back reaction potential, generically Λ br ( h = 0) may not be vanishing, and thenone needs Λ ( h = v ) (cid:29) Λ ( h = 0) . (8)Also, in order not to destabilize the weak scale size of the Higgs VEV, its magnitudeshould be bounded as Λ br ( h = v ) (cid:46) O ( v ) . (9)3n immediate consequence of the above conditions is that the relaxion should experi-ence a field excursion much bigger than f in the limit M h (cid:29) v :∆ φf (cid:38) M h v . (10)The required excursion is huge in the case that the back reaction potential is generatedby the QCD anomaly, in which Λ ∼ f π m π and therefore∆ φf (cid:38) O (cid:18) M h f π m π (cid:19) ∼ (cid:18) M h v (cid:19) . (11)Even when the scale of the back reaction potential saturates the bound (9), the requiredrelaxion excursion is still much larger than f as long as M h is higher than the weak scaleby more than a few orders of magnitudes. Note that the natural size of M h is the cutoffscale of the model for non-SUSY case, while it is the soft SUSY breaking scalar mass forSUSY case.Therefore, in the relaxion scenario, the hierarchy M h /v (cid:29) φ/f (cid:38) M h /v . Although ∆ φ (cid:29) f might be stable against radiativecorrections, it is still crying for an explanation with a sensible UV completion. To incorpo-rate a huge relaxion excursion, one may simply assume that the relaxion is a non-compactfield variable. See [3–10] for recent discussions of the related issues. In this paper, wediscuss an alternative scenario in which the relaxion corresponds to an exponentially longmulti-helical flat direction in the compact field space spanned by N sub-Planckian periodicaxions: φ i ≡ φ i + 2 πf i ( i = 1 , , .., N )with f i (cid:28) M Planck . Such a long flat direction is formed by a series of mass mixing betweennearby axions, producing a multiplicative sequence of helical windings of flat direction,which results in ∆ φf i = O ( e ξN )for ξ = O (1). Our scenario is inspired by the recent generalization of the axion alignmentmechanism for natural inflation [11] to the case of N axions [12]. Although it requires a4ather specific form of axion mass mixings, our scheme does not involve any fine tuningof continuous parameters, nor an unreasonably large discrete parameter.As we will see, our scheme finds a natural UV completion in high scale or (mini) splitsupersymmetry (SUSY) scenario with soft SUSY breaking scalar mass m SUSY (cid:29) v . Inthe UV completed model, the axion scales are generated by SUSY breaking [13–15] as f i ∼ (cid:112) m SUSY M ∗ , where M ∗ can be identified as the Planck scale or the GUT scale. With the ( N − f i to generate the desiredaxion mass mixings, the canonically normalized relaxion has a field range∆ φ ≡ πf eff ∼ πf i (cid:32) N − (cid:89) j =1 n j (cid:33) , where n j > j -th hidden sector. One can then arrange the microscopic parameters in a technicallynatural way to make the resulting relaxion potential V ( φ ) and the φ -dependent Higgsboson mass µ h ( φ ) vary with an exponentially large periodicity of O ( f eff ), while the backreaction potential V br ( h, φ ) has a periodicity of O ( f i ). An interesting feature of our modelis that the desired V ( φ ) and µ h ( φ ) arise as a natural consequence of the solution of theMSSM µ -problem advocated in [13–15].The outline of the paper is as follows. In the next section, we describe the basic ideawith a simple toy model and discuss the scheme within the framework of an effectivetheory of N axions. In section 3, we present a UV model with high scale SUSY, realizingour scheme in the low energy limit. Section 4 is the conclusion. II. EXPONENTIALLY LONG RELAXION FROM MULTIPLE AXIONS
To illustrate the basic idea, let us begin with a simple two axion model. The lagrangiandensity of the model is given by L = 12 ( ∂ µ φ ) + 12 ( ∂ µ φ ) − (cid:16) ˜ V + V + µ h | h | + V br + ... (cid:17) , (12)5here h is the SM Higgs doublet and φ i are the periodic axions: φ i ≡ φ i + 2 πf i , (13)with a scalar potential ˜ V = − Λ cos (cid:18) φ f + n φ f (cid:19) ,V = − (cid:15)f cos (cid:18) φ f + δ (cid:19) ,µ h = M h − (cid:15) (cid:48) f cos (cid:18) φ f + δ (cid:48) (cid:19) ,V br = − Λ ( h ) cos (cid:18) φ f + δ (cid:19) , (14)where Λ (cid:29) (cid:15)f (cid:29) Λ . (15)Here M h is an axion-independent mass parameter which is comparable to the cutoff scaleof the above effective lagrangian, and n > (cid:15)f (cid:38) O ( (cid:15) (cid:48) M h ) , (cid:15) (cid:48) f (cid:38) O ( M h ) , (16)and therefore the model is stable against the radiative corrections which replace the Higgsoperator | h | with the cutoff-square of O ( M h ), while allowing µ h = 0 for certain value of φ .As for the back reaction potential, one can consider two different possibilities. Oneoption is to generate it by the coupling of φ to the QCD anomaly, yieldingΛ ( h ) ∼ y u Λ h, (17)where y u denotes the up quark Yukawa coupling to the SM Higgs field h , and Λ QCD is the QCD scale. This option corresponds to the minimal model, however genericallyis in conflict with the axion solution to the strong CP problem. Alternative option is6o introduce a new hidden gauge interaction which confines around the weak scale andgenerates a back reaction potential given by [1, 16]Λ = m | h | + m (18)with m < m v (cid:46) O ( v ) . (19)In order for the model to be technically natural, the underlying dynamics to generate theback reaction potential should be arranged to make sure that the above conditions on m and m are stable against radiative corrections.The above two axion model involves the shift symmetries U (1) i : φ i f i → φ i f i + c i ( i = 1 , , (20)which are broken by ˜ V down to the relaxion shift symmetry U (1) φ : φ f → φ f + nc, φ f → φ f − c. (21)The flat direction associated with U (1) φ has a helical winding structure in the compact2-dim field space of φ i as depicted in Fig. (1). Then the periodicity of the flat directionis enlarged as ∆ φ = 2 π (cid:113) n f + f ≡ πf eff , (22)which is larger than the original axion periodicities 2 πf ∼ πf by the winding number n . The relaxion shift symmetry U (1) φ is slightly broken by small nonzero values of (cid:15), (cid:15) (cid:48) and Λ br . Note that this particular form of U (1) φ breaking is technically natural as longas the first condition of (16) is satisfied. To find the effective potential of the flat relaxiondirection, one can rewrite the model in terms of the canonically normalized heavy andlight axions [11, 12]: φ H = f φ + nf φ f eff , φ = nf φ − f φ f eff , (23)7 ⇢ = 12 ˙ + M g = g M H (3 + ✏ ) ✏ (1 e N ✏ ) + M g (92)( & M /g ) (93) ⇠ H f M Pl (94) g = ⇤ /M ( f = M ) (95) M . ✏ / ⇤ M Pl / , ⇤ = 1 GeV250 TeV , ⇤ = 100 GeV for ✏ = 10 (96) N &
50 log ✓ M ⇤ ◆ ⇡ ⇥ (97) H = ⇤ (98) (99)2 ⇡f /n (100)12 ⇢ = 12 ˙ + M g = g M H (3 + ✏ ) ✏ (1 e N ✏ ) + M g (92)( & M /g ) (93) ⇠ H f M Pl (94) g = ⇤ /M ( f = M ) (95) M . ✏ / ⇤ M Pl / , ⇤ = 1 GeV250 TeV , ⇤ = 100 GeV for ✏ = 10 (96) N &
50 log ✓ M ⇤ ◆ ⇡ ⇥ (97) H = ⇤ (98) (99)2 ⇡f /n (100)
13 2 ⇡ f / n n ( ) ⇡ f ( ) ⇡ f ( ) ⇡ f ( ) ··· ⇡f /n n (101)2 ⇡f (102)2 ⇡f (103)2 ⇡f (104) · · · ⇡f /n n (101)2 ⇡f (102)2 ⇡f (103)2 ⇡f (104) · · · ⇡f /n n (101)2 ⇡f (102)2 ⇡f (103)2 ⇡f (104) · · · n (105)2 ⇡f /n (106) φ φ · · · FIG. 1: Flat direction in the fundamental domain of axion fields in the limit Λ = 0. Eventhough the fundamental domain is sub-Planckian with f i ≪ M P l , the flat direction can have asuper-Planckian length if one (or both) of n i / gcd ( n , n ) is large enough. The right panel depictsthe flat direction in the fundamental domain for which the axion periodicity is manifest. which can be identified as the inflaton direction. One easily finds that the length of thisperiodic flat direction is given by ℓ flat = 2 π ! n f + n f gcd ( n , n ) , (12)where gcd ( n , n ) denotes the greatest common divisor of n and n . This shows thata super-Planckian flat direction with ℓ flat ≫ M P l ≫ f i can be developed on the two-dimensional sub-Planckian domain if n gcd ( n , n ) or n gcd ( n , n ) ≫ M P l f i ≫ . (13)In Fig. 1, we depict the flat direction in the fundamental domain of axion fields, which hasa length given by (12). Since the axionic inflaton of natural inflation rolls down along thisperiodic flat direction, its effective decay constant is bounded as f eff ≤ ℓ flat π , which means that at least one of n i should be as large as gcd ( n , n ) f eff /f i .Turning on the second axion potential∆ V = Λ " − cos m φ f + m φ f $% , (14)a nontrivial potential is developed along the periodic flat direction having a length (12).Even when ℓ flat ≫ M P l , natural inflation is not guaranteed as the inflaton potential induced6
FIG. 1: Flat relaxion direction in the two axion model. for which φ f = n φf eff + f n f + f φ H f H φ f = − φf eff + nf n f + f φ H f H , (24)where f H = f f /f eff . In the limit Λ (cid:29) (cid:15)f (cid:29) Λ , it is straightforward to integrate outthe heavy axion φ H to derive the low energy effective lagrangian of the light axion φ . Theresulting effective potential of the canonically normalized φ is given by V eff = − (cid:15)f cos (cid:18) φf eff − δ (cid:19) + (cid:18) M h − (cid:15) (cid:48) f cos (cid:18) φf eff − δ (cid:48) (cid:19)(cid:19) | h | − Λ ( h ) cos (cid:18) φf + δ (cid:19) , (25)where f eff = (cid:113) n f + f ≡ nf. (26)We can now generalize the above two axion model to the case of N > L = 12 (cid:88) i ( ∂ µ φ i ) − (cid:16) ˜ V + V + µ h | h | + V br + ... (cid:17) , (27)8here ˜ V = − N − (cid:88) i =1 Λ i cos (cid:18) φ i f i + n i φ i +1 f i +1 (cid:19) V = − (cid:15)f N cos (cid:18) φ N f N + δ N (cid:19) ,µ h = M h − (cid:15) (cid:48) f N cos (cid:18) φ N f N + δ (cid:48) N (cid:19) ,V br = − Λ ( h ) cos (cid:18) φ f + δ (cid:19) , (28)with Λ i (cid:29) (cid:15)f N (cid:29) Λ . The model involves the N axionic shift symmetries: U (1) i : φ i f i → φ i f i + c i (29)which are broken by ˜ V down to the relaxion shift symmetry: U (1) φ : φ i f i → φ i f i + γ i c ( γ i = − n i γ i +1 ) , (30)with the corresponding flat direction given by φ ∝ N (cid:88) i =1 ( − i − (cid:32) N − (cid:89) j = i n j (cid:33) f i φ i . (31)Turing on small nonzero values of (cid:15), (cid:15) (cid:48) and Λ br , the relaxion shift symmetry (30) isslightly broken and nontrivial potential of φ is developed. Although our way to break U (1) φ is rather specific, it is technically natural as the model involves many continuous ordiscrete axionic shift symmetries which are distinguishing our particular way of symmetrybreaking from other possibilities. It is again straightforward to integrate out the ( N − V [12]. For the canonically normalized φ ,the resulting effective potential is given by V eff = V ( φ ) + µ h ( φ ) | h | + V br ( h, φ )= − (cid:15)f N cos (cid:18) φf eff + ( − ) N − δ N (cid:19) + (cid:18) M h − (cid:15) (cid:48) f N cos (cid:18) φf eff + ( − ) N − δ (cid:48) N (cid:19)(cid:19) | h | − Λ ( h ) cos (cid:18) φf + δ (cid:19) , (32)9here f eff = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 (cid:32) N − (cid:89) j = i n j (cid:33) f i ∼ (cid:32) N − (cid:89) j =1 n j (cid:33) f ∼ e ξN f ( ξ = O (1)) ,f = f eff (cid:16)(cid:81) N − j =1 n j (cid:17) ∼ f . (33)For simplicity here we assumed that all f i are comparable to each other, or f is thebiggest among { f i } .Obviously, in the limit N (cid:29) f eff and another effectivedecay constant f which is comparable to the original decay constants f i . Such a bigdifference between f eff and f can be understood by noting that in order for the N -th axion φ N to travel one period along the relaxion direction, i.e. ∆ φ N = 2 πf N , the other axions φ i ( i = 1 , , ..., N −
1) should experience a multiple windings as ∆ φ i = 2 π (cid:16)(cid:81) N − j = i n j (cid:17) f i .This results in φ i f i = ( − i − (cid:32) N − (cid:89) j = i n j (cid:33) φf eff + ..., (34)where the ellipsis stands for the ( N −
1) heavy axions receiving a large mass from ˜ V . Foran illustration of this feature, we depict in Fig. (2) the relaxion field direction for the caseof N = 3 , n = 2 , n = 4.The effective potential (32) can easily realize the relaxion mechanism under severalconsistency conditions. First of all, like (16) of the two axion model, we need (cid:15)f N (cid:38) O ( (cid:15) (cid:48) M h ) , (cid:15) (cid:48) f N (cid:38) O ( M h ) , (35)in order for the model to be stable against radiative corrections, while allowing µ h = 0 forcertain value of φ . Without invoking any fine tuning, there is always a certain range of δ N and δ (cid:48) N for which the relaxion rolls down toward the minimum of V ( φ ) starting froma generic initial value φ with µ h ( φ ) = O ( M h ) >
0. After a field excursion ∆ φ = O ( f eff ),the relaxion is crossing µ h ( φ ) = 0, and then a nonzero Higgs VEV is developed together10 ⇢ = 12 ˙ + M g = g M H (3 + ✏ ) ✏ (1 e N ✏ ) + M g (92)( & M /g ) (93) ⇠ H f M Pl (94) g = ⇤ /M ( f = M ) (95) M . ✏ / ⇤ M Pl / , ⇤ = 1 GeV250 TeV , ⇤ = 100 GeV for ✏ = 10 (96) N &
50 log ✓ M ⇤ ◆ ⇡ ⇥ (97) H = ⇤ (98) (99)2 ⇡f /n (100) 12 ⇢ = 12 ˙ + M g = g M H (3 + ✏ ) ✏ (1 e N ✏ ) + M g (92)( & M /g ) (93) ⇠ H f M Pl (94) g = ⇤ /M ( f = M ) (95) M . ✏ / ⇤ M Pl / , ⇤ = 1 GeV250 TeV , ⇤ = 100 GeV for ✏ = 10 (96) N &
50 log ✓ M ⇤ ◆ ⇡ ⇥ (97) H = ⇤ (98) (99)2 ⇡f /n (100) ⇢ = 12 ˙ + M g = g M H (3 + ✏ ) ✏ (1 e N ✏ ) + M g (92)( & M /g ) (93) ⇠ H f M Pl (94) g = ⇤ /M ( f = M ) (95) M . ✏ / ⇤ M Pl / , ⇤ = 1 GeV250 TeV , ⇤ = 100 GeV for ✏ = 10 (96) N &
50 log ✓ M ⇤ ◆ ⇡ ⇥ (97) H = ⇤ (98) (99)2 ⇡f /n (100) ⇡f /n n (101)2 ⇡f (102)2 ⇡f (103)2 ⇡f (104) ⇡f /n n (101)2 ⇡f (102)2 ⇡f (103)2 ⇡f (104) ⇡f /n n (101)2 ⇡f (102)2 ⇡f (103)2 ⇡f (104) ⇢ = 12 ˙ + M g = g M H (3 + ✏ ) ✏ (1 e N ✏ ) + M g (92)( & M /g ) (93) ⇠ H f M Pl (94) g = ⇤ /M ( f = M ) (95) M . ✏ / ⇤ M Pl / , ⇤ = 1 GeV250 TeV , ⇤ = 100 GeV for ✏ = 10 (96) N &
50 log ✓ M ⇤ ◆ ⇡ ⇥ (97) H = ⇤ (98) (99)2 ⇡f /n (100) ⇡f /n n (101)2 ⇡f (102)2 ⇡f (103)2 ⇡f (104) · · · n (105)2 ⇡f /n (106) FIG. 2: Flat relaxion direction in the three axion case with n = 2 and n = 4. with the back reaction potential stabilizing the relaxion at the value giving (cid:104) h (cid:105) = v . Thestabilization condition leads to (cid:15)f N f eff ∼ Λ ( h = v ) f . (36)From (35), this then yields a lower bound on f eff : f eff f (cid:38) M h Λ ( h = v ) = (cid:18) M h v (cid:19) v Λ ( h = v ) , (37)where v / Λ ( h = v ) ∼ when V br is generated by the QCD anomaly, or v / Λ ( h = v )has a model-dependent value not exceeding O (1) when V br is generated by the hidden colordynamics which confines around the weak scale.To summarize, in our scheme for the relaxion mechanism, v (cid:28) M h can be technicallynatural with an exponential hierarchy between the two effective axion scales: f eff f = O ( e ξN ) ( ξ = O (1)) (38)11hich is arising as a consequence of a series of mass mixing between nearby axions inthe compact field space of N axions. Although it relies on a rather specific form of axionmass mixings, the scheme does not involve any fine tuning of continuous parameters, noran unreasonably large discrete parameter. III. A UV MODEL WITH HIGH SCALE SUPERSYMMETRY
In this section, we construct an explicit UV completion of the N axion model discussedin the previous section. We first note that our scheme requires that the axion potential˜ V should dominate over the other part of the potential in (28) as it determines the keyfeature of the model, i.e. an exponentially long flat direction in the compact field spaceof N axions. Specifically we need f i (cid:29) | ˜ V | (cid:29) | V | (cid:38) M h . (39)On the other hand, we wish to have an explicit UV model providing the full part of theaxion potential in (28), as well as a mechanism to generate the axion scales f i . Thisimplies that our UV model should allow the natural size of the Higgs boson mass, i.e. M h , to be much lower than its cutoff scale. As SUSY provides a natural framework forthis purpose, in the following we present a supersymmetric UV completion of the lowenergy effective potential (32).First of all, to have N axions with the decay constants f i (cid:28) M Planck , we introduce N global U (1) symmetries under which U (1) i : X i → e iβ i X i , Y i → e − iβ i Y i ( i = 1 , , ..., N ) , (40)where X i and Y i are gauge-singlet chiral superfields with the U (1) i -invariant superpoten-tial W = (cid:88) i X i Y i M ∗ , (41)where M ∗ corresponds to the cutoff scale of the model, which might be identified as theGUT scale or the Planck scale. Here and in the following, we ignore the dimensionless12oefficients of order unity in the lagrangian. We assume that SUSY is softly broken withSUSY breaking soft masses m SUSY ∼ M h (cid:28) M ∗ . (42)In particular, the model involves the soft SUSY breaking terms of X i and Y i , given by L soft = − m X i | X i | − m Y i | Y i | + (cid:18) A i X i Y i M ∗ + h . c (cid:19) , (43)where m X i ∼ m Y i ∼ A i ∼ m SUSY . To achieve the N axions in the low energy limit, we need all m X i are tachyonic, whichresults in (cid:104) X i (cid:105) ≡ x i ∼ (cid:112) m SUSY M ∗ , (cid:104) Y i (cid:105) ≡ y i ∼ (cid:112) m SUSY M ∗ . (44)Then the canonically normalized axion components φ i can be identified as X i ∝ e iφ i /f i , Y i ∝ e − iφ i /f i (45)with f i = (cid:113) x i + 9 y i ) ∼ (cid:112) m SUSY M ∗ . (46)Now we need a dynamics to generate the axion potential ˜ V in (28), developing anexponentially long flat direction as described in the previous section. For this purpose,we introduce ( N −
1) hidden Yang-Mills sectors associated with the gauge group G = (cid:81) N − i =1 SU ( k i ), including also the charged matter fieldsΨ i + Ψ ci , Φ ia + Φ cia ( i = 1 , , ..., N − a = 1 , , ..., n i ) , (47)where Ψ i and Φ ia are the fundamental representation of SU ( k i ), while Ψ ci and Φ cia are anti-fundamentals. These gauged charged matter fields couple to the U (1) i -breaking fields X i through the superpotential W = N − (cid:88) i =1 ( X i Ψ i Ψ ci + X i +1 Φ ia Φ cia ) . (48)13ote that X i couples to a single flavor of the SU ( k i )-charged hidden quark Ψ i + Ψ ci , while X i +1 couples to n i flavors of the SU ( k i )-charged hidden quarks Φ ia + Φ cia . With this formof hidden Yang-Mills sectors, the N global U (1) symmetries are explicitly broken downto a single U (1) by the U (1) i × SU ( k j ) × SU ( k j ) anomalies. The charged matter fieldsΨ i + Ψ ci and Φ ia + Φ ia get a heavy mass of O ( f i ), so can be integrated out at scales below f i . This yields an axion-dependent threshold correction to the holomorphic gauge kineticfunction τ i of SU ( k i ) at scales below f i : τ i = 1 g i + i π (cid:18) φ i f i + n i φ i +1 f i +1 (cid:19) + θ M λ i , (49)where we ignored the dependence on | X i | , while including the soft SUSY breaking by thegaugino masses M λ i ∼ m SUSY . As a consequence, at scales below f i , the global symmetrybreaking by the U (1) i × SU ( k j ) × SU ( k j ) anomalies is described by the following axioneffective interactions: N − (cid:88) i =1 π (cid:18) φ i f i + n i φ i +1 f i +1 (cid:19) (cid:16) F ˜ F (cid:17) SU ( k i ) , (50)where ( F ) SU ( k i ) denotes the gauge field strength of SU ( k i ) and ( ˜ F ) SU ( k i ) is its dual. As wewish to generate the axion potential | ˜ V | (cid:29) M h ∼ m from the above axion couplings,we assume ˜Λ i (cid:29) m SUSY , (51)where ˜Λ i denotes the confining scale of the hidden gauge group SU ( k i ). In such case, theresulting axion potential is determined by the non-perturbative effective superpotentialdescribing the formation of the SU ( k i ) gaugino condensation [17]: W np ∼ (cid:104) λ i λ i (cid:105) ∝ (cid:16) e − π τ i (cid:17) /k i , (52)yielding ˜ V = − N − (cid:88) i =1 Λ i cos (cid:18) k i (cid:18) φ i f i + n i φ i +1 f i +1 (cid:19)(cid:19) (53)14ith Λ i ∼ π k i M λ i ˜Λ i . (54)Our next mission is to generate the axion potential V and the axion-dependent Higgsmass-square µ h in (28), driving the evolution of the relaxion during the inflationary epoch,while scanning the Higgs mass-square from an initial value of O ( m ) to zero. Thiscan be done by introducing a superpotential term given by W = (cid:18) X N − M ∗ + X N M ∗ (cid:19) H u H d , (55)together with the associated K¨ahler potential term:∆ K = X N − X ∗ N M ∗ + h . c . (56)Here we ignore the irrelevant terms such as | X N | or | X N − | in the K¨ahler potential. Notethat the couplings in W leads to a logarithmically divergent radiative correction to ∆ K [18], and our model is stable against such radiative correction as long as the coefficient of∆ K is of order unity. Note also that W provides a solution to the MSSM µ problem asit yields naturally the Higgsino mass µ eff ∼ m SUSY [13–15].After integrating out the ( N −
1) axions which receive a heavy mass from ˜ V , whileleaving the light relaxion φ as described in the previous section, the K¨ahler potential term∆ K gives rise to V = − m cos (cid:18) n N − + 1) φf eff + δ (cid:19) , (57)where m ∼ f N − f N M ∗ m ∼ m ,f eff = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 (cid:32) N − (cid:89) j = i n j (cid:33) f i ∼ (cid:32) N − (cid:89) j =1 n j (cid:33) f , (58)and δ is a phase angle which is generically of order unity. In our scheme, the MSSMHiggsino mass µ eff originates from W , and therefore is naturally of the order of m SUSY N −
1) heavy axions, we find the MSSM Higgsparameters µ eff and Bµ eff depend on the relaxion φ as µ eff = µ N − exp( − i n N − φ/f eff ) + µ N exp( i φ/f eff ) ,Bµ eff = b N − exp( − i n N − φ/f eff ) + b N exp( i φ/f eff ) , (59)where | µ N | ∼ | µ N − | ∼ f M ∗ ∼ m SUSY , | b N | ∼ | b N − | ∼ m . (60)Then the determinant of the MSSM Higgs mass matrix D = ( m H u + | µ eff | )( m H d + | µ eff | ) − | Bµ eff | (61)also depends on φ via | µ eff | = | µ N | + | µ N − | + 2 | µ N µ N − | cos (cid:18) n N − + 1) φf eff + δ µ N − δ µ N − (cid:19) , | Bµ eff | = | b N | + | b N − | + 2 | b N b N − | cos (cid:18) n N − + 1) φf eff + δ b N − δ b N − (cid:19) , (62)where δ µ and δ b are the phases of µ and b , respectively. Obviously, for an appropriaterange of δ µ and δ b , the determinant D can flip its sign from positive to negative as therelaxion experiences an excursion of O ( f eff ). Once the relaxion is stabilized near the pointof D = 0, the MSSM Higgs doublets H u and H d can be decomposed into the light SMHiggs h with a mass of O ( v ) and the other heavy Higgs bosons having a mass of the orderof m SUSY (cid:29) v .To complete the model, we need to generate the back reaction potential V br . In regardto this, we simply adopt the schemes suggested in [1]. One option is to generate V br through the QCD anomaly. For this, one can introduce W br = X QQ c , (63)where Q + Q c is an exotic quark which receive a heavy mass by (cid:104) X (cid:105) ∼ f . Once thisheavy quark is integrated out, the axion φ couples to the gluons as132 π φ f (cid:16) F ˜ F (cid:17) QCD . (64)16fter the ( N −
1) heavy axions are integrated out, this leads to the relaxion-gluon coupling132 π φf (cid:16) F ˜ F (cid:17) QCD , (65)where f = f eff (cid:16)(cid:81) N − j =1 n j (cid:17) ∼ f . (66)Then the resulting back reaction potential is obtained to be V br ( h, φ ) ≈ − y u Λ h cos (cid:18) φf + δ br (cid:19) , (67)where y u is the up quark Yukawa coupling to the SM Higgs field h , and δ br is a phaseangle of order unity.Alternatively, we can consider a back reaction potential generated by an SU ( n HC )hidden color gauge interaction which confines at scales around the weak scale [1, 16]. Forthis, one can introduce the hidden colored matter superfields L + L c , N + N c (68)with the superpotential couplings W br = κ X M ∗ LL c + κ u H u LN c + κ d H d L c N, (69)where L is an SU ( n HC )-fundamental and SU (2) L -doublet with the U (1) Y charge 1/2, L c is its conjugate representation, N is an SU ( n HC )-fundamental but SU (2) L × U (1) Y singlet, and N c is its conjugate representation. At scales below m SUSY , all superpartnerscan be integrated out, leaving the following Yukawa interactions between the relevantlight degrees of freedom: L br = m L e iφ /f LL c + κ u sin βhLN c + κ d cos βh † L c N + m N N N c , (70)where L + L c and N + N c denote the fermion components of the original superfields,tan β = (cid:104) H u (cid:105) / (cid:104) H d (cid:105) , and m L ∼ κ f M ∗ ∼ κ m SUSY (71)17s presumed to be lighter than m SUSY . Note that a nonzero Dirac mass of N + N c isinduced by radiative corrections below m SUSY , giving m N ∼ π sin(2 β ) κ u κ d m ∗ L e − iφ /f ln (cid:18) m SUSY m L (cid:19) . (72)Now this effective theory at scales below m SUSY corresponds to the non-QCD modelproposed in [1, 16], yielding a back reaction potential of the form V br = m hh † cos (cid:18) φf + δ (cid:19) + m cos (cid:18) φf + δ (cid:19) , (73)where we have expressed the axion component φ in terms of the light relaxion field φ ,and m ∼ κ u κ d sin(2 β ) m L Λ , m ∼ m N Λ (74)for the SU ( n HC ) confinement scale Λ HC . If m < m v with m (cid:46) O ( v ), which canbe achieved for m L < πv [1], this back reaction potential can successfully stabilize therelaxion at a value giving v = (cid:104) h (cid:105) (cid:28) m SUSY . IV. CONCLUSION
In this paper, we have addressed the problem of huge scale hierarchy in the relaxionmechanism, i.e. a relaxion excursion ∆ φ ∼ πf eff which is bigger than the period 2 πf of the back reaction potential by many orders of magnitudes. We proposed a schemeto yield an exponentially long relaxion direction within the compact field space of N periodic axions with decay constants well below the Planck scale, giving f eff /f ∼ e ξN with ξ = O (1). Although it relies on a specific form of the mass mixing between nearbyaxions, our scheme does not involve any fine tuning of continuous parameters, nor anunreasonably large discrete parameter. Furthermore, our scheme finds a natural UVcompletion in high scale or (mini) split SUSY scenario, in which all decay constants ofthe N periodic axions are generated by SUSY breaking as f i ∼ √ m SUSY M ∗ , where m SUSY denotes the soft SUSY breaking scalar masses and M ∗ is the fundamental scale such asthe Planck scale or the GUT scale. In our model, the required relaxion potential and18he relaxion-dependent Higgs boson mass are generated through a superpotential termproviding a natural solution to the MSSM µ -problem. V. ACKNOWLEDGMENT
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