Kaluza-Klein Theories
Kaluza-Klein theories • Add extra space-time • The 5-D general coordinate dimensions to unify gravity invariance broken in ground and electromagnetism state ➝ ordinary gravity in (T. Kaluza 1921) 4-D plus an Abelian gauge field • The fifth dimension is • U(1) gauge symmetry appears compact, periodic, very small associated with coordinate (O. Klein 1926) transformations on circle • Start with a theory of Einstein • Parameters of the two gravity in 5-D. One of the theories connected, since they dimensions is compactified in have same origin a very small circle • Tower of massive scalars
M 2 n/R2 KK / 113 Kaluza theory • Consider a 5D theory only with gravity, the gravitons are
hMN M,N = µ, 5
• Correspond to fluctuations around flat space
gMN = ⌘MN + hMN
which decompose into a spin 2 h µ ⌫ , identified with a graviton, a spin 0, h µ 5 identified with a photon, and a scalar h55 • The first proposal did not include massless charged matter particles
114 faster as r increases, F 1/r2+d, than the gauge forces, F 1/r2. Of course, we know that at ⇠ ⇠ very large distances gravity lives in 4D, since we know that Newton’s law reproduces very accurately, for example, the orbits of the planets. This means that the extra dimensions must be compact with a compactification radius R. At distances larger than R we will have a 4D theory with Newton’s law: m m F = G 1 2 , (52) N r2
where GN is the observed Newton’s constant. Matching Eqs. (51) and (52) at r = R one gets G G = grav . (53) N Rd
Therefore large R implies a small GN . In other words, 4D gravity must be weaker than the other interac- tions if its field lines spread over large extra dimensions. The larger the extra dimensions, the weaker is gravity. This is a very interesting possibility that, as we will see below, has spectacular phenomenological implications. Several years later Randall and Sundrum found a different reason to have extra dimensions [43]. If the extra dimensions were curved or ‘warped’, gravitons would behave differently than gauge bosons and this could explain their different couplings to matter. Below we will discuss these two scenarios in more detail. Let us first explain the situation in the old Kaluza–Klein picture.
9 Kaluza–Klein theories As we said before, Kaluza was one of the first to consider theories with more than four dimensions in an attempt to unify gravity with electromagnetism. Klein developed this idea in 1926 using a formalism that is usually called Kaluza–Klein reduction [44]. Although their initial motivation and ideas do not seem to be viable, the formalism that they and others developed is still useful nowadays. This is the one that will be considered below.
y
xµ
Kaluza-KleinFig. 14: Compactification theories on S1
For• simplicity,The action we will in start 5D, with y is a 5Dfifth-dimension. field theory of scalars. Only The scalars action is given by
4 2 2 2 4 S5 = d xdyM @µ + @y + g5 , (54) ⇤ | | | | | | Z h i where by y we refer to the extra fifth dimension.y = Wey have+2⇡ extractedR a universal scale M in front of the y compactified in a circle ⇤ action in•This order corresponds to keep the to the 5D identification field with the of samey with mass-dimensiony +2⇡R. In such as a in case, 4D. we Let can us expand now consider the 5D complex that the y y +2⇡R Thisfifth corresponds dimensionscalar field is to in compact the Fourier identification and series: flat. We of willwith consider that. it In has such the a topology case, we of can a circle expandS1 theas in 5D Fig. complex14. scalar field• inExpand Fourier series: the complex scalar in a Fourier series 1 iny/R (n) (0) iny/R (n) (x, y)= e 25(x)= (x)+ e (x) , (55) 1 iny/R (n) (0) iny/R (n) (x, y)= ne= (x)= (x)+ n=0e (x) , X 1 X6 (55) n= n=0 that inserted in Eq. (54) andX 1 integrated over y gives X6 that inserted inIntegrating Eq. (54) and over integrated y gives over y gives (0) (n) • S5 = S4 + S4 (56) (0) (n) S = S + S 2/r (56) masslesswhere scalar 5 4 4 1/r (0) 4 (0) 2 2 (0) 4 0 where S4 = d x 2⇡RM @µ + g5 , (57) ⇤ | | | | Z h i (0) 2 1 (n) 4 4 (0) 2 (2n) 2(0) 4 nFig. 17: The(n Kaluza) 2 Klein tower of massive states due to an extra S dimension. Masses mn = n /r grow S = d x 2⇡RM @ + g , | | 4 S4 = d x 2⇡RMµ @µ 5 + linearly with the fifth dimension’s+ quartic wave number n Z. couplings . (58)(57) ⇤ | ⇤ | | | | | R | | 2 Z Z n=0 h X6 ⇣i ⌘ 4 µ µ µ tower of 2 =2⇡ r d x @ ' (x ) @ ' (x )⇤ + ... =2⇡r + ... (n) 4 (n) 2 n (n) 2 0 µ 0 S4D Z massiveS 4modes= d x 2⇡RM @µ + + quartic couplings(0) . (58) We see that the above action⇤ corresponds to115 a 4D theoryThis with means that a the massless 5D action reduces scalar to one 4D action forand a massless a scalar tower field plus of an infinite sum | | R of| massive scalar| actions in 4D. If we are only interested in energies smaller than the 1 scale, we may Z (n) n(0)=0 (n) r massive modes . The field X6 will be referred to⇣ as⌘ theconcentrate zero-mode, only on the action while of the massless mode.will be referred to as Kaluza–Klein (KK) modes. 3.2 Compactification of a Vector Field in 5 Dimensions(0) Vector fields are decomposed in a completely analogous way: AM = Aµ,A4 = . Consider the We see that the above action corresponds to a 4D theory with a massless scalar and{ } a{ tower} of action This(n) reduction of a(0) 5D theory to a 4D theory allows one to treat 5D theories(n) as1 4D field theories. = d5x F F MN (46) massive modes . The field will be referred to as the zero-mode, whileS5 D willg2 MN be referred to as This is very useful since we know much more about 4D theories than 5D theories.Z At5D low energies (large with a field strength Kaluza–Klein (KK) modes. F = @ A @ A (47) distances) we know that massive states in 4D theories can be neglected. ThereforeMN M theN N effectiveM theory implying (0) Thisat energies reduction below of a1 5D/R is theory described to a by 4D the theory zero-mode allows Eq. one (57 to). Aftertreat 5Dnormalizing theories@M @ A @ asM @ 4D,A we=0 field. obtain theories. from (48) M N N M This is veryEq. useful (57) since we know much more about 4D theoriesIf we now than choose 5D a gauge, theories. e.g. the transverse gauge: At low energies (large M M (0) @ AM =0,A0 =0 @ @M AN =0, (49) 4 (0) 2 2 (0) 4 ) distances) we know that massive statesS4 in= 4D theoriesd x @µ can be+ neglected.g4 , Therefore the effective theory(59) | | then this obviously| becomes| equivalent to the scalar(0) field case (for each component AM ) indicating an at energies below 1/R is described by the zero-modeZ Eq. (57infinite). After tower of massive normalizing states for each massless state in 5D., weIn order obtain to find the 4D effective from action we h once again plug this intoi the 5D action: Eq. (57) where the 4D self-coupling is given by 2 S5D 7! S4D g 2⇡r 2⇡r 2 5 = d4x F µ⌫ F + @ ⇢ @µ⇢ + ... . (0) g = . 2 (0) (0)µ⌫ 2 µ 0 0 4 4 (0) 2 2 (0) 4 g5D g5D (60) S4 = d x @µ 2⇡RM+ g4 , Z ✓ ◆ (59) F µ⌫ ⇢ | | ⇤ Therefore| we end| up with a 4D theory of a massless gauge particle (0), a massless scalar 0 from the massless Kaluza-Klein state of and infinite towers of massive vector and scalar fields. Notice that the This equation tells us that the strengthZ of theh interaction of the zero-modei decreases as the radiusMN in-µ⌫ where the 4D self-coupling is given by gauge couplings of 4- and 5 dimensional actions (coefficients of FMNF and Fµ⌫F ) are related by creases. If R is large, the scalar is weakly coupled. 1 2⇡r 2 2 = 2 . (50) g g4 g5 The general features described aboveg2 = for a 5D5 scalar. will also hold for gauge fields and gravity.(60) 4 2⇡RM 24 After Kaluza–Klein reduction, we will have a 4D theory⇤ with a massless gauge field and a graviton: This equation tells us that the strength of the interaction2 of the zero-mode decreases as the radius in- 4 M 1 MN creases. If R is large, the scalar is weaklyd xdyM coupled.⇤ + 2 F FMN ⇤ 2 R 4g5 The general features describedZ above for a 5D scalar will also hold for gauge fields and gravity. 4 2 (0) 2⇡R (0) µ⌫ (0) After Kaluza–Klein reduction, we= will haved xM a 4D⇡RM theory with+ a massless2 F F gaugeµ⌫ + field... and a graviton: ⇤ ⇤ R 4g Z 5 4 M 21 (0)1 1 (0) µ⌫ (0) 4 d x + MN2 F Fµ⌫ + ... , (61) d⌘ xdyM 16⇤⇡GN+R F4g FMN Z ⇤ 2 R 4g2 4 Z 5 where (0) is the 4D scalar-curvature containing the zero-mode2⇡R (massless) graviton, and F (0) is the R 4 2 (0) (0) µ⌫ (0) = d xM ⇡RM + 2 F Fµ⌫ + ... gauge field-strength of the zero-mode⇤ (massless)⇤ R gauge boson.4g From Eq. (61) we read Z 5 1 21 4 2(0) g5 (0) µ⌫ (0) d x g4 = + 2 F, Fµ⌫ + ... , (62)(61) ⌘ 16⇡GN R 2⇡RM4g4 Z ⇤ for(0) the 4D gauge coupling, and (0) where is the 4D scalar-curvature containing the zero-mode1 (massless) graviton, and F is the R G = , (63) gauge field-strength of the zero-mode (massless)N gauge16⇡ boson.2RM 3 From Eq. (61) we read ⇤ g2 g2 = 265 , (62) 4 2⇡RM ⇤ for the 4D gauge coupling, and 1 G = , (63) N 16⇡2RM 3 ⇤ 26 This corresponds to the identification of y with y +2⇡R. In such a case, we can expand the 5D complex scalar field in Fourier series:
1 (x, y)= einy/R (n)(x)= (0)(x)+ einy/R (n)(x) , (55) n= n=0 X 1 X6
This correspondsthat to the inserted identification in Eq. of (y54with) andy +2 integrated⇡RThis. In such corresponds over a case,y gives we can to the expand identification the 5D complex of y with y +2⇡R. In such a case, we can expand the 5D complex scalar field in Fourier series: scalar field in Fourier series: (0) (n) S5 = S4 + S4 (56) This corresponds to the identification1 iny/R (n of) y with(0) y +2⇡R.iny/R In such(n) a case, we can1 expand the 5D complex (x, y)= e (x)= (x)+ e (x(x,) , y)= (55)einy/R (n)(x)= (0)(x)+ einy/R (n)(x) , (55) scalar field inwhere Fourier series:n= n=0 X 1 X6 n= n=0 X 1 X6 that inserted in Eq. (54) and integrated(0) over1 y gives4 (0) 2 2 (0) 4 S4 = diny/Rx 2⇡RMthat(n) inserted@µ (0) in Eq.+ g5 (54 ) andiny/R integrated, (n) over y gives (57) (x, y)= e (0) (n()⇤x)=| |(x)+| e| (x) , (55) SZ5 = S + S (56) n= 4 4 h n=0 i2 (n) X 1 4 (n) 2 X6 n (n) 2 (0) (n) S4 = d x 2⇡RM @µ + + quarticS5 = S4 couplings+ S4 . (58) (56) where ⇤ | | R | | that inserted in Eq. (54) and integratedZ over y givesn=0 X6 ⇣ ⌘ (0) 4 (0) 2 2 (0)where4 S4 = d x 2⇡RM @µ + g5 , (57) ⇤ | | | | (0) (n) (0) We seeZ that the above actionS corresponds= S + toS a 4D theory with a massless scalar and a tower of h 5 i2 4 4 (56) (n) 4 (n) (n) 2 (0)n (n) 2(0) 4 (0) 2 (n) 2 (0) 4 S4 =massived modesx 2⇡RM . The@µ field+ will beS4 referred+ quartic= to ascouplings thed x zero-mode,2⇡RM. (58)@µ while + gwill5 be referred, to as (57) ⇤ | | R | | ⇤ Z n=0 | | | | where Kaluza–Klein (KK)X6 modes. ⇣ ⌘ Z h n i2 (n) (0)4 (n) 2 (n) 2 We see that(0) the above actionThis reduction corresponds of to a a 5D 4D theory with to a aS 4D massless4 theory= scalar allows d x oneand2⇡ aRM to tower treat of 5D theories@µ as+ 4D field theories.+ quartic couplings . (58) (n) 4(0) (0) 2 2 (0) 4 (n) ⇤ | | R | | massive modesS4 This=. The is very field d usefulx 2will⇡RM since be referred we@µ know to as the much+ zero-mode,g5 more about while, 4DZwill theories be referred than to 5Dn as=0 theories. At low(57) energies (large ⇤ | | | | X6 ⇣ ⌘ Kaluza–Klein (KK)distances) modes.Z we know that massive states in 4D theories can be neglected. Therefore the effective theory h n i2 This reduction(n) of a 5D theory4 to a 4D theory allowsWe( onen see) 2 to that treat the 5D theories above(n) as2 action 4D field corresponds theories. to a 4D(0) theory with a massless scalar (0) and a tower of S4 at= energies belowd x 2⇡1RM/R is described@µ by the+ zero-mode Eq. (+57 quartic). After normalizingcouplings . , we(58) obtain from This is very useful since we know much more⇤ about 4D| massive theories| than modes 5DR theories. (n| ). The At| low field energies (0) (largewill be referred to as the zero-mode, while (n) will be referred to as Eq. (57) Z n=0 ⇣ ⌘ distances) we know that massive states in 4DX theories6 Kaluza–Klein can be neglected. (KK) Therefore modes. the effective theory at energies below 1/R is described by the zero-mode Eq.(0) (57). After normalizing4 (0) (0)2, we obtain2 (0) from4 S4 = d x @µ + g4 , (0) (59) WeEq. (57 see) that the above action corresponds to a 4DThis theory reduction with| a of massless| a 5D theory| scalar| to a 4Dand theory a tower allows of one to treat 5D theories as 4D field theories. (n) (0) Z h (n)i massive modes . The field(0) will4 be referred(0) 2 2 to(0) as4 the zero-mode, while will be referred to as where the 4DS4 self-coupling= d x @ isµ givenThis+ by isg4 very useful, since we know much(59) more about 4D theories than 5D theories. At low energies (large | | | | 2 Kaluza–Klein (KK) modes. Z h distances) wei2 know thatg5 massive states in 4D theories can be neglected. Therefore the effective theory where the 4D self-coupling is given by g4 = . (60) (0) This reduction of a 5D theory to a 4Dg2at theory energies allows below one21⇡/R toRM treatis described 5D theories by the as 4D zero-mode field theories. Eq. (57). After normalizing , we obtain from g2 = 5 . ⇤ (60) This is very usefulThis equation since we tells know us much that4 the more2⇡RM strengthEq. about (57) 4D of the theories interaction than 5D of theories. the zero-mode At low decreases energies (large as the radius in- ⇤ distances)This equation we tells know us that that the massive strength of states the interaction in 4D theories of the zero-mode can be decreases neglected. as the Therefore radius(0) in- the effective4 (0) theory2 2 (0) 4 creases. If R is large, the scalar is weakly coupled. S4 = d x @µ + g4 , (59) creases. If R is large, the1/R scalar is weakly coupled. (0) | | | | at energies below Theis general describedWe features by can the described zero-modedescribe above Eq. 5D for (57 atheories). 5D After scalar normalizing will as also 4D hold onesZ for, we gaugeh obtain fields from and gravity.i The general features described• above for a 5D scalar will also hold for gauge fields and gravity. Eq. (57) After Kaluza–Klein reduction, wewhere will the have 4D a self-coupling4D theory with is a given massless by gauge field and a graviton: After Kaluza–Klein reduction, we will have a 4D theory with a massless gauge field and a graviton: 2 (0) 4 (0) 2 2 (0) 4 2 g5 S4 = d x @µ +2 g4 , g4 = (59). (60) M 2 1 | M| | 1 | 2⇡RM d4xdyM + 4F MNF MN ⇤ Z d2xdyMh MN ⇤ + 2 Fi FMN ⇤ ⇤ 2 R 4g5 ⇤ 2 R 4g where the 4D self-couplingZ is given by Z This equation tells us that5 the strength of the interaction of the zero-mode decreases as the radius in- 4 2 (0)creases.2⇡R If(0)g2Rµ⌫ is(0) large, the scalar2⇡R is weakly coupled. = d xM ⇡RM +2 4 2 F 5 Fµ⌫ 2+ ...(0) (0) µ⌫ (0) Same⇤ =for⇤ R ggauge4 d=4xMg5 fields⇡RM. and +graviton:2 F Fµ⌫ + ... (60) Z• 2⇡RM⇤ ⇤ R 4g 1 Z 1 The general features described5 above for a 5D scalar will also hold for gauge fields and gravity. d4x (0) + F (0) µ⌫F (0)⇤ + ... , After2 Kaluza–Kleinµ1⌫ reduction,1 we will(61) have a 4D theory with a massless gauge field and a graviton: This equation tells us⌘ that the strength16⇡GN R of the4g interaction44 of the(0) zero-mode(0) µ decreases⌫ (0) as the radius in- Z d x + 2 F Fµ⌫ + ... , (61) ⌘ 16⇡GN R 4g creases.(0) If R is large, the scalar is weakly coupled. 4 (0) 2 where is the 4D scalar-curvature containing theZ zero-mode (massless) graviton, and F 4 is the M 1 MN RThe general features described above for a 5D scalar will also hold ford xdyM gauge fields⇤ and+ gravity.F FMN gauge field-strengthwhere of the zero-mode(0) is the (massless) 4D scalar-curvature gauge boson. From containing Eq. (61) we the read zero-mode (massless)⇤ 2 graviton,R 4g and2 F (0) is the After Kaluza–KleinR reduction, we will have a 4D theory with a masslessZ gauge field and a graviton:5 gauge field-strength of the zero-modeg2 (massless) gauge boson. From Eq. (61) we read 2⇡R g2 = 5 , g4 is 4D4(62) gauge coupling2 (0) (0) µ⌫ (0) 4 = d xM ⇡RM + 2 F Fµ⌫ + ... 2⇡RM 2 ⇤ ⇤ R 4g 4 M⇤ 1 MN 2 GN is 4D Newton constant 5 d xdyM ⇤ + F2 FMNg5 Z for the 4D gauge coupling, and ⇤ 2 R 4g2g4 = , 1 1 (62) 1 5 2⇡RM d4x (0) + F (0) µ⌫F (0) + ... , (61) Z G = , 2 µ⌫ N 2 3 strength⇤ ⌘ of interaction(63) 16⇡G NsuppressedR 4g by 4 16⇡ RM 2 (0) 2⇡R (0) µ⌫ (0)Z 4 for the 4D gauge= coupling,d xM and⇡RM⇤ + 2 F Fradiusµ⌫ + of... extra dimension ⇤ ⇤ R (0) 4g5 (0) Z 26 where is the 4D1 scalar-curvature containing the zero-mode (massless) graviton, and F is the R GN = , (63) 4 1gauge(0) field-strength1 16(0)⇡ ofµ2⌫RM the(0) zero-mode3 (massless) gauge boson. From Eq. (61) we read d x + 2 F Fµ⌫⇤ + ... , (61) ⌘ 16⇡GN R 4g4 Z 116 g2 26 g2 = 5 , (62) where (0) is the 4D scalar-curvature containing the zero-mode (massless) graviton, and4 F2(0)⇡RMis the R ⇤ gauge field-strength of the zero-mode (massless) gauge boson. From Eq. (61) we read for the 4D gauge coupling, and 2 1 2 g5 GN = , (63) g4 = , 16⇡2RM(62)3 2⇡RM ⇤ ⇤ for the 4D gauge coupling, and 26 1 G = , (63) N 16⇡2RM 3 ⇤ 26 for the 4D Newton constant. Again, as in Eq. (60), the strength of the interaction is suppressed by the length of the extra dimension. Let us now imagine that we live in 5D. From Eqs. (62) and (63) we learn the following. Since the 2 2 gauge couplings g4 = (1) and g5 . 1 (in order to have a perturbative theory) we have from Eq. (62) IfO we lived in 5D, then g5 should be perturbative, but from the that • 4D perspective g4 should be strongly1 interacting ➝ R . (64) ⇠ M ⇤ On the other hand, using the relation G 1/R(8⇡M1/M2 ), where M =2.4 1018 GeV is from now on N ⌘ ⇠ P ⇤ P ⇥ the reduced Planck scale, we have from Eq. (63) that
2 3 MP =2⇡RM . (65) ⇤ Equations (64) and (65 ) imply
1 32 R = lP 10 cm . (66) ⇠ MP ⇠ We have then reached• The the conclusioncompactification that ifradius we live is of in the 5D, order the radius of the ofPlanck the extralength dimension must be of order the Planck length(reducedlP ! This Planck extra mass) dimension will not be accessible to present or near-future experi- ments. This is the reason why experimentalists never paid attention to the existence of extra dimensions even though they were motivated theoretically a long time ago, e.g., from string theory. Let us finish this section with a comment on117 the scale M . Classically, we introduced this scale ⇤ based on dimensional grounds. At the quantum level, however, this scale has a similar meaning as MP in 4D gravity or 1/pGF in Fermi theory. It represents the cutoff ⇤ of the 5D theory. We do not know how to quantize the 5D theory above M , since amplitudes such as grow with the energy as ⇤ ! E/M . ⇠ ⇤
10 Large extra dimensions for gravity In 1998 Arkani-Hamed, Dimopoulos, and Dvali (ADD) proposed a different scenario for extra dimen- sions [42]. Motivated by the weakness of gravity, they considered that only gravity was propagating in the extra dimension. As we already saw, the effective 5D theory at distances larger than R is a theory of 4D gravity with a GN being suppressed by the length of the extra dimension. Then the smallness of GN can be considered a consequence of large extra dimensions. The key point to avoid the conclusion of Eq. (66) is that not all fields should share the same dimensions. In particular, gauge bosons should be localized in a 4D manifold. In 1995 string theorists realized that superstrings in the strong-coupling limit contain new solitonic solutions [45]. These solutions received the name of D-branes and consisted in sub-manifolds of dimen- sions D+1 (less than 10) with gauge theories living on them. From string theory we therefore learn that there can be theories where gravitons and gauge bosons do not share the same number of dimensions, giving realizations of the scenario proposed by ADD [46]. Let us then assume that gravity lives in more dimensions than the SM particles (leptons, quarks, the Higgs and gauge bosons), and study the implications of this scenario. First of all, we must find out how large the extra dimensions must be in order to reproduce the right value of GN . For d flat and compact extra dimensions, we have
2 2 4 d d M 4 d d M (0) d xd yM ⇤ = d xV M ⇤ + ... , (67) ⇤ 2 R ⇤ 2 R Z Z where V d is the volume of the extra dimensions. Hence we have
2 d 2+d MP = V M . (68) ⇤
27 • Gauge coupling in 5D has negative mass dimension ➝ non-renormalizable From 4D perspective this is due to the K-K modes accesible at the energy scale
• M is the cut-off of the theory, which we treat as an effective⇤ field theory below this mass scale
118 In D space time dimensions, this generalizes to
1 VD 4 2 = 2 (51) g4 gD
where Vn is the volume of the n dimensional compact space (e.g. an n sphere of radius r).
3.2.1 The electric (and gravitational) potential We apply Gauss’ law for the electric field E~ and the potential of a point charge Q:
1 1 E~ dS~ = Q E~ , :4D · )k k/R2 / R SI2 1 1 E~ dS~ = Q E~ , :5D · )k k/R3 / R2 In D space time dimensions, this generalizes to SI3
1 VThus,D 4 the apparent behaviour of the force depends upon whether we are sensitive to the extra dimension 2 = or2 not: if we test the force at distances smaller than(51) its size (i.e. at energies high enough to probe such g4 gD small distance scales), it falls off as 1/R3: the field lines have an extra dimension to travel in. If we test 2 where Vn is the volume of the n dimensional compactthe space force (e.g. at larger an n distancessphere of than radius the sizer). of the extra dimension, we obtain the usual 1/R law. • Gauss’ law for the electricIn Dfieldspace and time dimensionspotential of a point charge 3.2.1 The electric (and gravitational) potential 1 1 E~ , . (52) ~ D 2 D 3 We apply Gauss’ law for the electric field E and the potential of a point charge Q: k k/R / R 1 If one dimension is compactified (radius r) like in M4 S , then we have two limits 1 1 ⇥ E~ dS~ = Q E~ , :4D · )k k/R2 / R 1 I2 3 : R
3.2.2 Sketch of Compactified Gravitation 25
The spin 2~ graviton GMN becomes the 4D graviton gµ⌫, some gravivectors Gµn and some graviscalars Gmn (where m, n =4,...,D 1), along with their infinite Kaluza-Klein towers. The Planck mass 2 D 2 D 2 D 4 squared MPl = M VD 4 M r is a derived quantity. Fixing D, we can fix MD and r to get D ⇠ D the correct result for M 1019 GeV. So far, we require M > 1 TeV and r<10 16cm from Standard Pl ⇠ D Model measurements since no significant confirmed signature of extra dimensions has been seen at the time of writing.
3.3 Brane Worlds In the brane world scenario, we are trapped on a 3+1 surface in a D +1dimensional bulk space-time (see Fig. 18). There are two cases here: large extra dimensions and warped space-times. Since gravity itself is so weak, the constraints on brane world scenarios are quite weak: the extra dimension is constrained to 16 be of a size r<0.1 mm or so, potentially much larger than the 10 cm of the Standard Model, hence the name large extra dimensions.
25 for the 4D Newton constant. Again, as in Eq. (60), the strength of the interaction is suppressed by the length of the extra dimension. Let us now imagine that we live in 5D. From Eqs. (62) and (63) we learn the following. Since the gauge couplings g2 = (1) and g2 1 (in order to have a perturbative theory) we have from Eq. (62) 4 O 5 . that 1 R . (64) ⇠ M ⇤ On the other hand, using the relation G 1/(8⇡M 2 ), where M =2.4 1018 GeV is from now on N ⌘ P P ⇥ the reduced Planck scale, we have from Eq. (63) that
2 3 MP =2⇡RM . (65) ⇤ Equations (64) and (65) imply 1 32 R = lP 10 cm . (66) ⇠ MP ⇠ We have then reached the conclusion that if we live in 5D, the radius of the extra dimension must be of order the Planck length lP ! This extra dimension will not be accessible to present or near-future experi- ments. This is the reason why experimentalists never paid attention to the existence of extra dimensions even though they were motivated theoretically a long time ago, e.g., from string theory. Let us finish this section with a comment on the scale M . Classically, we introduced this scale ⇤ based on dimensional grounds. At the quantum level, however, this scale has a similar meaning as MP in 4D gravity or 1/pGF in Fermi theory. It represents the cutoff ⇤ of the 5D theory. We do not know how to quantize the 5D theory above M , since amplitudes such as grow with the energy as ⇤ ! E/M . ⇠ ⇤
10 Large extra dimensions for gravity In 1998 Arkani-Hamed, Dimopoulos, and Dvali (ADD) proposed a different scenario for extra dimen- sions [42]. Motivated by the weakness of gravity, they considered that only gravity was propagating in the extra dimension. As we already saw, the effective 5D theory at distances larger than R is a theory of 4D gravity with a GN being suppressed by the length of the extra dimension. Then the smallness of GN can be considered a consequence of large extra dimensions. The key point to avoid the conclusion of Eq. (66) is that not all fields should share the same dimensions. In particular, gauge bosons should be localized in a 4D manifold. In 1995 string theorists realized that superstrings in the strong-coupling limit contain new solitonic solutions [45]. These solutions received the name of D-branes and consisted in sub-manifolds of dimen- sions D+1 (less than 10) with gauge theories living on them. From string theory we therefore learn that there can be theories whereLarge gravitons and extra gauge bosons dimensions do not share the same number of dimensions, giving realizations of the scenario proposed by ADD [46]. Let• us thenSimilarly assume for that the gravity gravitational lives in more fields dimensions than the SM particles (leptons, quarks, the Higgs andG gaugeMN in bosons), 5D becomes and study the the graviton implications gμν, gravivectors of this scenario. Gμ Firstn and of all, we must find out how large thegraviscalars extra dimensions Gmn must be in order to reproduce the right value of GN . For d flat and compact extra dimensions, we have Is it possible to solve the hierarchy problem with large extra • 2 2 dimensions? 4 d d M 4 d d M (0) d xd yM ⇤ = d xV M ⇤ + ... , (67) ⇤ 2 R ⇤ 2 R • The PlanckZ mass is now a derivedZ quantity, in d dimensions, V is the where V d is thevolume volume of the extra dimensions. Hence we have d d For a toroidal compactification we have V =(22 ⇡dR) 2+. Followingd Ref. [42], we will absorb the factors 11 MP = V M . (68) 2⇡ in M and rewrite Eq. (68) as ⇤ ⇤ 2 d 2 MP =(RM ) M . (69) ⇤ ⇤ Note that Eq. (64) does not apply since gauge bosons27 do not live in 5D. Let us fix M slightly above ⇤ the electroweak• Gravity scale M lives TeVin more to avoid dimensions introducing than a new the scalegauge (this bosons, is a nullification it of the hierarchy ⇤ ⇠ problem). In suchpropagates a case we in have the fromextra Eq. dimensions, (69) a prediction thus explaining for R: the gravitational couplings smallness compared to the other couplings d =1 R 109 km , ! ⇠ d =2 R 0.5 mm , ! 120 ⇠ . . d =6 R 1/(8 MeV) , ! ⇠ The option d =1is clearly ruled out. For d =2we expect changes in Newton’s law at distances below the mm. Surprisingly, as we will show below, we have not measured gravity at distances below 0.1 mm. This is due to the fact that Van der Waals forces become comparable to gravity at distances ⇠ around 1 mm, making it very difficult to disentangle gravity effects from the large Van der Waals effects. So the option d =2is being tested today at the present experiments. Larger values of d are definitely allowed.
10.1 Phenomenological implications What are the implications of this scenario? Let us concentrate on the case d =2. At distances shorter than 1 mm, we must notice that gravity lives in 6D. To study the effects of a 6D gravity, we will again Fourier decompose the 6D graviton field, hµ⌫(x, y1,y2). For example, if y1 and y2 are compactified in a torus, we have the Fourier decomposition
1 1 i(n1y1+n2y2)/R (~n) hµ⌫(x, y1,y2)= e hµ⌫ (x) , (70) n1= n2= X 1 X 1 (~0) (~n) where ~n =(n1,n2). The state hµ⌫ is our massless graviton, while hµ⌫ with ~n =0are the KK states 2 2 62 2 that, from a 4D point of view, are massive particles of masses m~n =(n1 + n2)/R . Therefore we can describe this 6D theory as a 4D theory containing a massless graviton and a KK tower of graviton states. There are also the KK states for the components hµ5, hµ6, h65, h55, and h66. Nevertheless, since matter is assumed to be confined in a 4D manifold at y =0, we have that the energy-momentum µ ⌫ tensor has only 4D components, TMN = ⌘M ⌘N Tµ⌫ (y). Hence these extra states do not couple to the energy-momentum tensor of matter. The situation is a little bit more subtle for the ‘dilaton’ field that corresponds to a combination of hMN M,N =5, 6. Although it does not couple to Tµ⌫, it mixes with the graviton. This mixing can be eliminated by a Weyl transformation. Nevertheless, after the Weyl rotation, appears to be coupled to the trace of Tµ⌫. This coupling is usually smaller than those between gravitons and matter (in fact, it is zero for conformal theories) and therefore we will neglect it. The effective Lagrangian for the KK gravitons, after normalizing the kinetic term of the gravitons is given by
1 2 ( ~n)µ⌫ (~n) ( ~n)µ (~n)⌫ 1 (~n) µ⌫ KK = kin m~n h h µ⌫ h µh ⌫ + h Tµ⌫ , (71) L L 2 MP ~n=0 X6 ⇣ ⌘ (~n) where is the kinetic term of the gravitons. The KK states hµ⌫ will modify the gravitational in- Lkin teraction at E>1/R. Since they couple to matter with a strength 1/M , we have that at energies ⇠ P 11 2 6 2 4 In string theory, where Mst plays the role of M , we have for d =6that MP =2⇡(RMst) Mst/g4 . ⇤
28 E>1/R, the (dimensionless) gravitational strength squared grows as
ER ER E2 E2 E 4 g2 (ER)2 , (72) grav ⇠ M 2 ⇠ M 2 ⇠ M n =0 n =0 P P X1 X2 ✓ ⇤ ◆ 2 where in the last equality we have used Eq. (69). Note that ggrav becomes (1) at energies M . There- O ⇤ fore M is the scale at which quantum gravity effects are important. The generalization to d extra ⇤ dimensions is given by E 2+d g2 . (73) grav ⇠ M ✓ ⇤ ◆ With Eq. (72) we can easily estimate any gravitational effect in any experimental process that we can imagine.
E>1/R, the (dimensionless) gravitational strength squared grows as Fig. 15: Upper limits on forces of the form of Eq. (74)[47] ER ER E2 E2 E 4 g2 (ER)2 , (72) grav ⇠ M 2 ⇠ M 2 ⇠ M n =0 n =0 P P X1 X2 ✓ ⇤ ◆ 2 where in the last equality we have used Eq. (69). Note that ggrav becomes (1) at energies M . There- O ⇤ fore M is the scale at which quantum gravity effects are important. The generalization to d extra 10.1.1 Measuring the gravitational force at millimetre distances⇤ dimensions is given by E 2+d The KK of the graviton give rise to new forces. Since they are massive particles they produceg2 a Yukawa-. (73) KK tower of graviton ➝ grav ⇠ M type force. For the first KK• (n = 1,n =0and n =0,n = 1) of masses 1/R, this✓ force⇤ ◆ is given new1 ± forces2 1 With2 Eq.± (72) we can easily estimate any gravitational effect in any experimental process that we can by imagine. m1m2 r/ FKK(r)= ↵GN e , (74) r where ↵ = 16/3 for a 2-torus compactification and = R. Searches for new forces have been carried out at several experiments. Nevertheless, the bounds on ↵ are very weak at distances r below 0.1 mm. ⇠ In Fig. 15 we plot the presentλ experimental= R, α constant bounds on ↵ofand . The value of R 0.5 mm, expected ⇠ for M TeV and d =2, is ruled out. Therefore the case d =2is only at present allowed if M & 3 ⇤ ⇠ compactification type ⇤ TeV [47]. • If we assume M* above 1TeV: 29 • For d=1 ➝ R~ 109 km ➝ d=2 R~ 0.5 mm Fig. 15: Upper limits on forces of the form of Eq. (74)[47] d = 2 is ruled out, unless • 10.1.1 Measuring the gravitational force at millimetre distances The KK of the graviton give rise to new forces. Since they are massive particles they produce a Yukawa- M* > 3 TeV type force. For the first KK (n = 1,n =0and n =0,n = 1) of masses 1/R, this force is given E. G. Adelberger,1 J. ±H. Gundlach,2 B. R. Heckel1 et al.,2 Prog.± Part. Nucl. Phys. 62 (2009) 102–134. by m1m2 r/ F (r)= ↵G e , (74) KK N r where ↵ = 16/3 for a 2-torus compactification and = R. Searches for new forces have been carried out at several experiments. Nevertheless, the bounds on ↵ are very weak at distances r below 0.1 mm. 121 ⇠ In Fig. 15 we plot the present experimental bounds on ↵ and . The value of R 0.5 mm, expected ⇠ for M TeV and d =2, is ruled out. Therefore the case d =2is only at present allowed if M & 3 ⇤ ⇠ ⇤ TeV [47].
29 Superstrings and branes more symmetries, more unification, more particles, more dimensions • Particles are excitations of • Duality relates strongly vibrating strings coupled to weakly coupled theories • Branes are p-dimensional surfaces in space time • Could be ultimate unification • Many require SUSY • Require extra space-time dimensions for consistency • Similar idea as K-K compactification • Many ways to compactify these extra dimensions • Spectrum includes a graviton 122 • Adding more dimensions we extend the Poincaré symmetries of SM and general coordinate transformations of general relativity, to each
• Superstring theory also requires extra dimensions 10 for heterotic string, 11 for supergravity and M theory
• Why don’t we perceive them?
• They are compactified or
Fig. 16: Picture of different extra-dimensional set-ups: the brane (on the left), where in string theory SM states appear as open strings whose ends end upon the brane but gravitons appear as close string states in the bulk, or compactification (on the right), in this example we have taken the example of a circle S1 times ordinary 4- dimensional Minkowski space M 4. Fig. 18: Force field lines feel the effect of the extra dimensions. Here we show a 3+1 dimensional brane, where gravity spreadsWe into the extralive dimension on and feels a its brane effect. — bulk of space time lives in D+3+1 dimensions, but • 3.1 Compactification and a Scalar Field in 5 Dimensions 3.3.1 LargeSM extra dimensions(us) are fixed to a braneTaking = compactified 3+1 dimensional extra dimensions as an example,hyper consider surface. a massless five dimensional (5D) scalar There is the possibility to try to solve the hierarchy problem with the large extra dimensionsfield (i.e. scenario a scalar if field living in a 5-dimensional bulk space-time) '(xM ),M =0, 1,...,4 with action we put M 1 TeV. The idea is that this is the fundamental scale: there is no high scale associated with DGravity⇠ moves through space-time, i.e. all dimensions. We don’t. MPl fundamentally - it is an illusion caused by the presence of the extra dimensions. In 5D for example, 5 M D 2 8 = d x@ '@ '. (43) MPl2 = M VD 4 r 10 km, clearly ruled out by observations. Already in 6D though, r =0.1 5D M D ) ⇠ S mm - consistent with experiments that measure the gravitational force on small distance scales. This Z rephrases the hierarchy problem to the question “why are the extra dimensions so largeWe singlecompared the with extra dimension out by calling it x4 = y. y defines a circle of radius r with y y +2⇡r. 16 10 cm?” 1 ⌘ Our space time is now M4 S . Periodicity in the y direction implies that we may perform a discrete Graviton phenomenology: each Kaluza-Klein mode couples weakly 1/MPl, but there are so ⇥ / Fourier expansion many modes that after summing over them, you end up with 1/MD suppression only! One can approxi- 123 mate them by a continuum of modes with a cut-off. The graviton tower propagates into the bulk and takes µ 1 µ iny away missing momentum leading to a pp j + p~miss signature (for example) by the process shown in '(x ,y)= 'n(x )exp . (44) ! T r Fig. 19. n= ✓ ◆ X 1 Notice that the Fourier coefficients are functions of the standard 4D coordinates and therefore are (an 3.3.2 Warped (or ‘Randall-Sundrum’ space-times infinite number of) 4D scalar fields. The equations of motion for the Fourier modes are the (in general Warped space-times are where the metric exponentially warps along the extra dimensionmassive)y: Klein-Gordon wave equations 2 ky µ ⌫ 2 ds = e | |⌘µ⌫dx dx + dy . (54) 2 M 1 µ n µ iny The metric changes from y =0to y = ⇡r via ⌘ e k⇡r⌘ . Here, we set M = M , but this gets @ @M ' =0 @ @µ 2 'n(x )exp =0 µ⌫ 7! µ⌫ D Pl ) r r warped down to the weak brane: n= ✓ ◆ ✓ ◆ X 1 k⇡r 2 ⇤⇡ MPle (TeV), (55) n ⇠ ⇠ O µ µ µ = @ @µ'n(x ) 2 'n(x )=0. (45) if r 10/k. Here, k is of order M and so we have a small extra dimension, but the warping explains ) r ⇠ Pl the smallness of the weak scale. Note that we still have to stabilize the separationThese between are the branes, then an infinite number of Klein Gordon equations for massive 4D fields. This means that each which can involve extra tuning unless extra structure is added to the model. 2 n2 Fourier mode 'n is a 4D particle with mass m = 2 . Only the zero mode (n =0) is massless. One can The interaction Lagrangian is n r = Gµ⌫T /⇤ , visualize the(56) states as an infinite tower of massive states (with increasing mass proportional to n). This is LI µ⌫ ⇡ where T is the stress energy tensor, containing products of the other Standard Modelcalled fields. a Kaluza⇤ Klein tower and the massive states (n =0) are called Kaluza Klein-states or momentum µ⌫ ⇡ ⇠ 6 (TeV), so the interaction leads to electroweak-strength cross sections, not gravitationallystates, sincesuppressed they come from the momentum in the extra dimension: O ones. Thus, the LHC can produce the resonance: one will tend to produce the lightestIn one order most tooften, obtain as the effective action in 4D for all these particles, let us plug the mode expansion of ' 26 Eq. 44 into the original 5D action Eq. 43: 2 4 1 µ µ µ n 2 = d x dy @ ' (x ) @ ' (x )⇤ ' S5D n µ n r2 | n| n= ✓ ◆ Z Z X 1 23 124 Fig. 18: Force field lines feel the effect of the extra dimensions. Here we show a 3+1 dimensional brane, where gravity spreads into the extra dimension and feels its effect.
3.3.1Fig. Large 18: extraForce field dimensions lines feel the effect of the extra dimensions. Here we show a 3+1 dimensional brane, where gravity spreads into the extra dimension and feels its effect. There is the possibility to try to solve the hierarchy problem with the large extra dimensions scenario if we put MD 1 TeV. The idea is that this is the fundamental scale: there is no high scale associated with 3.3.1⇠ Large extra dimensions MPl fundamentally - it is an illusion caused by the presence of the extra dimensions. In 5D for example, ThereD is2 the possibility to try8 to solve the hierarchy problem with the large extra dimensions scenario if MPl2 = MD VD 4 r 10 km, clearly ruled out by observations. Already in 6D though, r =0.1 we put M 1 TeV.) The⇠ idea is that this is the fundamental scale: there is no high scale associated with mm - consistentD with⇠ experiments that measure the gravitational force on small distance scales. This MPl fundamentally - it is an illusion caused by the presence of the extra dimensions. In 5D for example, rephrases the hierarchyD 2 problem to8 the question “why are the extra dimensions so large compared with MPl2 = M VD 4 r 10 km, clearly ruled out by observations. Already in 6D though, r =0.1 16 D ) ⇠ 10 cm?”mm - consistent with experiments that measure the gravitational force on small distance scales. This Gravitonrephrases the phenomenology: hierarchy problem each to the Kaluza-Klein question “why mode are the couples extra dimensions weakly so large1/M comparedPl, but there with are so 16 10 cm?” / many modes that after summing over them, you end up with 1/MD suppression only! One can approxi- Graviton phenomenology:Warped each Kaluza-Klein extra dimensions mode couples weakly 1/M , but there are so mate them by a continuum of modes with a cut-off. The graviton tower propagates/ Pl into the bulk and takes 1/M away missingmany modes momentum that after leading summing to over a pp them, youj + endp~miss up withsignatureD (forsuppression example) only! by One the can process approxi- shown in mate them by a continuum of modes with! a cut-off.T The graviton tower propagates into the bulk and takes Fig. 19.away missing momentum leading to a pp j + p~miss signature (for example) by the process shown in ! T Fig. 19. • 5th dimension bounded by two (3+1) branes 3.3.2 Warped (or ‘Randall-Sundrum’ space-times 3.3.2 Warped (or• ‘Randall-Sundrum’Warped space-times, space-times the metric warps exponentially Warped space-times are wherealong thethe metricextra dimension exponentially warps along the extra dimension y: Warped space-times are where the metric exponentially warps along the extra dimension y: 2 ky µ ⌫ 2 ds =2 e | ky|⌘µ⌫dxµ dx⌫ + dy2 . (54) ds = e | |⌘µ⌫dx dx + dy . (54)
MD = MPlanck, but it gets warpedkk⇡ ⇡rdownr to eW scale, The metricThe metric changes changes from• fromy =0y =0to yto=y =⇡r⇡rviavia⌘⌘µµ⌫⌫ ee ⌘µ⌘⌫µ.⌫ Here,. Here, we we set M setD M= DMPl=,M butPl this, but gets this gets r~10/k, k~ Planck scale 7!7! small extra dimension warpedwarped down todown the to weak the weak brane: brane: ⟹
k⇡r ⇤⇡ MPle k⇡r (TeV), (55) ⇤⇡ ⇠M e ⇠ O (TeV), (55) ⇠ Pl ⇠ O if r 10/k. Here, k is of order MPl and so we have a small extra dimension, but the warping explains if r 10/k⇠. Here, k is of order M and so we have a small extra dimension, but the warping explains ⇠ the smallness of the weak scale.Pl Note that we still have to stabilize the separation between the branes, the smallnesswhich can of involve the weak extra scale. tuning Note unless that extra we structure still125 have is added to stabilize to the model. the separation between the branes, which can involveThe interaction extra tuning Lagrangian unless is extra structure is added to the model. = Gµ⌫T /⇤ , (56) The interaction Lagrangian is LI µ⌫ ⇡ µ⌫ where Tµ⌫ is the stress energy tensor, containing= G productsT /⇤ of the, other Standard Model fields. ⇤⇡ (56) I µ⌫ ⇡ ⇠ (TeV), so the interaction leads to electroweak-strengthL cross sections, not gravitationally suppressed where TO is the stress energy tensor, containing products of the other Standard Model fields. ⇤ ones.µ⌫ Thus, the LHC can produce the resonance: one will tend to produce the lightest one most often, as ⇡ ⇠ (TeV), so the interaction leads to electroweak-strength cross sections, not gravitationally suppressed O ones. Thus, the LHC can produce the resonance: one26 will tend to produce the lightest one most often, as
26 Planck brane is warped down to weak brane. Higgs and other fields live on the weak brane
Gravity may propagate in the extra dimension ⟹ might produce a TeV resonance that is detectable — “RS graviton”
126 Electricidad
Magnetismo Electromagnetismo
Luz Interacción electrodébil
Decaimiento beta Interacción débil Neutrinos ¿GUT? Modelo Estándar
¿Teoría Protones Unificada?
Neutrones Interacción fuerte ¿SUSY?
Piones
Gravitación Gravedad terrestre ¿Supercuerdas? universal Relatividad general Mecánica celestial Geometría del espacio-tiempo
127 Multi-Higgs models more particles
• 2HDM without SUSY • Complex MSSM ℂMSSM • widely studied • different versions • More sources of CP depending on how violation they couple to the other SM particles • Candidates for dark matter, with some • 3 or more HDM also discrete symmetry possible • Might give explanation • Extra singlet Higgs for the mass models NMSSM hierarchies and mixing of quarks and leptons
128 recent talk by Strocchi [174], where he reiterates the basic points exposed al length in his book [175].
3. Two-Higgs-doublet models 3.1. Several Higgs doublets: generic features Within the SM, the single Higgs doublet is overstretched. It takes care simultaneously of the masses of the gauge bosons and of the up and down-type fermions. N-Higgs-doublet models (NHDM), which are among the simplest extensions of the SM Higgs sector, relax this requirement. They are based on the simple suggestion that the notion of generations can be brought to the Higgs sector. Since the gauge structure of the SM does not restrict the number of Higgs “generations”, it needs to be established experimentally, and anticipating future experimental results, one can investigate this possibility theoretically. As a bonus, NHDMs lead to remarkably rich phenomenology. They allow for signals which are impossible within the SM, such as several Higgs bosons, charged and neutral, modification of the SM-like Higgs couplings, FCNC at tree level, additional forms of CP-violation from the scalar sector, and opportunities for cosmology such as scalar DM candidates and modification of the phase transitions in early Universe. Also, many bSM models including supersymmetry (SUSY), gauge unification models, andN-Higgs even string theory doublet constructions models naturally — lead NHDM to several Higgs doublets at the electroweak scale. The NHDM lagrangian• Add more uses NcomplexHiggs doublets scleras electroweaki (we now switch doublets to the small Greek letter for the doublets), i =1,...,NAll, allwith having same the hyper same charge hypercharge Y=1 Y =1.Thisechoesinthescalarpotentialand Yukawa interactions. The renormalizable Higgs self-interaction potential can generically be written as
V ( )=Yij i† j + Zijkl( i† j)( k† l) . (38)
The hermiticity of V N implies 2 + N 2 ( thatN 2 + the 1) / 2 coe realcients parameters: satisfy Yij = Yji⇤, Zijkl = Zjilk⇤ , Zklij = Zijkl.For • 2 2 2 N doublets, the general12 for potential 2HDM, contains 54 for N3HDM…+ N (N +1)/2 real parameters: 14 for 2HDM, 54 for 3HDM, etc. These parameters must also comply with the requirements that the potential be bounded from below to insure• Potential existence ofmust the globalbe bounded minimum by andbelow, that no the charge minimum or becolour neutral and not charge- breaking, which is anotherbreaking exotic minima possibility absent in the SM. The values of the quartic coe cients Zijkl cannot be too large in order not to overshoot the perturbative unitarity bounds [176, 177]. The Yukawa sector is given• Must by Nrespectcopies unitarity of Eq. (31): bounds ¯ ¯ ¯ ˜ Y = L⇧i` i + QL idR ➝i + QL iuR i + h.c. (39) • Can have L CP breaking minima baryogenesis (or disaster) where we suppressed the fermion flavor indices. In the most general case, all matrices ⇧i, i, i are independent, with the number of free parameters skyrocketing129 to 54N. Even though many of them can be removed by a scalar and fermion space basis change, the dimension of the physical parameter space remains huge. Once the scalar potential is written, the procedure for the SSB closely follows the SM. One minimizes the potential and performs the global symmetry transformation to bring the vevs to the following form:
1 u 1 0 1 = , i = ,i=2,...,N. (40) h i p2 v1 h i p2 vi ✓ ◆ ✓ ◆ In order for the vacuum to be neutral, not charge-breaking, u must be zero. There is no fine-tuning associated with this requirement; neutrality of the vacuum can be guaranteed if the coe cients in front of terms ( i† i)( j† j) ( i† j)( j† i)arepositive.Sincealldoubletscoupletothegauge-bosonsinthe 2 2 2 same way, the W and Z masses are determined by the single value v = v1 + + vN , and, just as in the SM, the ⇢-parameter remains equal to one at tree level. Inserting the scalar··· vevs| | in the Yukawa sector (39) produces the fermion mass matrices: 1 1 1 M = ⇧ v ,M= v ,M= v⇤ . (41) ` p i i d p i i u p i i 2 i 2 i 2 i X X X 18 2HDM • Most studied, experimental limits exist • 4 parameters in the quadratic part 10 parameters (6 real, 4 complex) in the quartic part • Again: minimum must be bounded from below, neutral and satisfy perturbative unitarity bounds The Higgs potential of the most general 2HDM is conventionally parametrized as 2 2 2 2 1 2 2 2 V = m11 1† 1 + m22 2† 2 m12 1† 2 (m12)⇤ 2† 1 + ( 1† 1) + ( 2† 2) 2 2 5 2 5⇤ 2 + ( † )( † )+ ( † )( † )+ ( † ) + ( † ) 3 1 1 2 2 4 1 2 2 1 2 1 2 2 2 1
+ 6( 1† 1)( 1† 2)+ 7( 2† 2)( 1† 2)+h.c. . (42) h i 2 2 2 It contains 4 independent free parameters in the quadratic part (real m11, m22 and complex m12)and 10 free parametersTo analyse in the quartic it: usually part (real impose 1,2,3,4 andsymmetries complex 5,6 ,7). Envisioning renormalization procedure,• [185, 186] argued that the most general 2HDM scalars sector should also include the o↵- µ diagonal kinetic terms {(Dµ 1)†(D 2)+h.c.. They can of course be eliminated by a non-unitary basis change at any given order of perturbation series so that does not enhance the space of physical 130 { possibilities. The parameters of potential (42) must satisfy stability constraints (the potential must be bounded from below), the neutrality of the vacuum, and, in order for the perturbative expansion to make sense and be reliable, perturbative unitarity constraints [176, 177]. Analyzing the most general 2HDM potential with the straightforward algebra is technically challenging, and in order to proceed further one either simplifies it by imposing symmetries or resorts to more elaborate mathematical methods.
3.2.1. Symmetries of the 2HDM scalar sector The number of free parameters can be reduced, and the model becomes analytically tractable, if one imposes additional global symmetries on the Higgs doublets. Anticipating the NFC principle for the Yukawa sector in 2HDM, one often considers the simplified 2HDM potential with 6 = 7 =0.The quartic part of this potential is then invariant under
Z2 : 1 1 , 2 2 , (43) ! ! 2 but the m12 term in the quadratic part, either real or complex, softly breaks this Z2 symmetry. With this simplified quartic potential, its stability in the strong sense [187] implies [188, 20]:
> 0 , > 0 , + > 0 , + + > 0 . (44) 1 2 1 2 3 1 2 3 4 | 5| 2 p p If in addition one sets m12 =0,theZ2 symmetry becomes exact; the inert doublet model (IDM) described in section 6.1.2 is based on such a choice. Furthermore, if 5 =0,themodelacquiresthe i↵ global U(1) symmetry 1 1, 2 e 2 known as Peccei-Quinn symmetry [189]. In total, there are six classes of family and!CP-type! symmetries which can be imposed on the Higgs sector of 2HDM without producing accidental symmetries [190, 191]. If one disregards U(1)Y gauge interactions, then even larger global basis changes are allowed, mixing i with ˜i. Within 2HDM, the full classification of such symmetries was given in [192, 193]; the phenomenology of the so called maximally symmetric 2HDM based on the softly broken SO(5) symmetry group was presented in [184].
3.2.2. Basis-invariant methods Coming back to the general 2HDM, we remark that not all of the 14 parameters in Eq. (42) are physical. If one focuses on the scalar sector only, then there exists a large freedom to change the basis in the ( 1, 2) space, leading to reparametrization of the Higgs potential (the explicit expressions can be found in [185, 194]). It allows one to remove three among 14 free parameters without a↵ecting the physical observables. The freedom of basis changes brings up the issue of hidden symmetries. The potential can be invariant under a certain non-obvious reflection in the ( 1, 2)space,whichafteranappropriatebasis change becomes the Z2-symmetry (43), but its presence is hard to detect in the original basis. In order
20 • Not all parameters are physical, only 11.
• Add more symmetries: No FCNC ➝ Z2 symmetry, softly 2 broken by m12 term
• Mainly two types of 2HDM analysed, 8 parameters:
Type I: all fermions couple only to one doublet Type II: up quarks couple to one doublet, down quarks and leptons couple to the second doublet
• After eW symmetry breaking there are five massive ± 2 physical Higgs bosons mh, m H, m A, m H , tanβ (v1/v2), m12 , sen(β-α) α mixing angle between scalars
131 Higgs doublets around minimum as
+ + '1 '2 1 = 1 , 2 = 1 . (51) (v1 + ⌘1 + i 1) (v2 + ⌘2 + i 2) ✓ p2 ◆ ✓ p2 ◆ 0 The Goldstone modes are G± = c '1± + s '2± and G = c 1 + s 2. The orthogonal combinations are the physical charged Higgs boson H± = s '1± + c '2±, and the neutral field ⌘3 = s 1 + c 2. In T general, the three neutral scalars ⌘i mix giving rise to the three neutral Higgs bosons: (H1,H2,H3) = T (⌘1, ⌘2, ⌘3) , with the rotation matrix parametrized with three angles ↵1, ↵2, ↵(n1,n2)3. For the original most basis in generalR 2HDM, the neutral mass matrix canR be written in di↵erent bases, see [214, 185]• and two di↵erent forms in section 5.10 and Appendix C of [20], as well as in the basis independent formwhich [198] and there with is natural the bilinear technique [190, 210]. Its explicit analytical diagonalization in terms of theflavour parameters conservation is cumbersome but still manageable [214]. (h1’,h2’) Higgs basis, all η Higgs basis • 2 the vev is in one of the h h′1 Higgses, rotated by β from original h′2 β H mass eigenstates
η (h,H) physical Higgs α 1 • bases, rotated by α
Figure 1: Three bases in the space of CP-even neutral scalar fields: the original basis (⌘1, ⌘2), the Higgsfrom basis original, (h10 ,h20 ), and by and the basis of mass eigenstates (H, h). (β-α) from Higgs one If the model is CP-conserving, then ⌘3 A decouples in the mass matrix from the first two and becomes the CP-odd pseudoscalar. The two⌘CP-even scalars mix and give rise to the mass eigenstates H = c↵⌘1 + s↵⌘2 and h = s↵⌘1 + c↵⌘2, with the convention that h is the ligher of the two. The mixing angle ↵ is usually defined from ⇡/2to⇡/2; alternative definitions are sometimes used. The relation among the three bases in the space of CP-even neutral fields is visualized132 in Fig. 1. The original basis (⌘1, ⌘2), in which the fermion sector featuring natural flavor conservation takes the simplest form, is rotated by angle to obtain the Higgs basis (h10 ,h20 ). In this basis, the vev is present only in the first doublet, so that the gauge bosons interact only with h10 but not h20 . The physical Higgses (H, h)form the third basis, rotated by ↵ with respect to the original one or by ↵ with respect to the Higgs hi basis. From here, one obtains the Higgs couplings to gauge bosons relative to the SM ⇠VV: ⇠h =sin( ↵) , ⇠H =cos( ↵) , ⇠A =0. (52) VV VV VV One immediately sees that 2HDMs, and in general NHDMs, predict that all such couplings are less hi 2 than one and they satisfy the following sum rule: i(⇠VV) =1[221].IntheCP-conserving case, the pseudoscalar boson A decouples from the gauge boson pairs at the tree level, which is easily seen in the P + Higgs basis. The same logic also leads to the absence of the tree-level coupling H W Z [222, 13]; such vertex can appear at tree level only in models with new scalars in higher-dimensional representations of SU(2)L. Finally, expressions for the trilinear and quartic scalar interactions are listed in [185, 20]. The alignment limit corresponds to ↵ ⇡/2. In this case, the lightest neutral h is SM-like ! Higgs and is identified with the experimentally observed h125. The second neutral Higgs H, even if it is relatively light, decouples from the gauge-boson pairs. In the other limit, ↵, the heavier state H is SM-like, and a second, lighter Higgs is predicted. ⇡
23 • How to test them? Look for signals of new physics • Usual test focus on 2HDM type I and II, and on MSSM • Heavy extra Higgs usually decoupled, difficult to discard the 2HDM
• Other new scalars would give similar results, so searches for new Higgses and scalars are complementary
• LEP excluded charge scalars MH >80 GeV, and b—>s� places stringer constraints MH>480 GeV
133 Toyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 3
may exclude the SM in favor of the 2HDM; they may exclude both; or they may be compatible with SM, thus restricting the 2HDM extension into a particular (eventually, small) region of the parameter space.
III. CP-CONSERVING MODEL
We start with the CP-conserving model and randomly generate points in the parameter space such that 2 2 2 2 mh = 125 GeV, 90 GeV mA 900 GeV, mh
Toyama International Workshop on Higgs as a Probe of New Physics 2013, 13–16, February, 2013 3 may exclude the SM in favor of the 2HDM; they may exclude both; or they may be compatible with SM, thus restricting the 2HDM extension into a particular (eventually, small) region of the parameter space.
III. CP-CONSERVING MODEL
We start with the CP-conserving model and randomly generate points in the parameter space such that 2 2 2 2 mh = 125 GeV, 90 GeV mA 900 GeV, mh
[272] A. Celis, V. Ilisie, A. Pich, LHC1 constraints on two-Higgs doublet models, JHEP 07 (2013) 053. RII,approx =sin2 (↵ ) . (3) ZZ arXiv:1302.4022 , doi:10.1007/JHEP07(2013)053tan2 ↵ tan2 . [273] C.-Y. Chen, S. Dawson, M. Sher, Heavy Higgs Searches and Constraints on Two Higgs Doublet Models, Phys. Rev. D88 (2013) 015018, [Erratum: Phys. Rev.D88,039901(2013)]. arXiv:1305. 1624, doi:10.1103/PhysRevD.88.015018,10.1103/PhysRevD.88.039901.
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FIG. 1: Points in the (sin ↵,tan ) plane that passed[278] J. Bernon, all J. the F. Gunion, constraints H. E. Haber, Y. in Jiang, type S. Kraml, I Scrutinizing (left) and the alignment in type limit in II two- (right) at 1 in Higgs-doublet models. II. mH=125 GeV, Phys. Rev. D93 (3) (2016) 035027. arXiv:1511.03682, green (light grey) and 2 in blue (dark grey). Alsodoi:10.1103/PhysRevD.93.035027 shown are the lines. for the SM-like limit, that is sin( ↵)=1 (negative sin ↵) and for the limit sin( + ↵)=1(positivesin↵). [279] X.-F. Han, L. Wang, Two-Higgs-doublet model of type-II confronted with the LHC run-I and run-II dataarXiv:1701.02678.
[280] G. Belanger, B. Dumont, U. Ellwanger, J. F. Gunion, S. Kraml, Global fit to Higgs signal strengths The points generated have then to pass all theand constraints couplings and implications previously for extended Higgs described sectors, Phys. Rev.plus D88 the (2013) combined075008. arXiv: [32] ATLAS 1306.2941, doi:10.1103/PhysRevD.88.075008. and CMS strengths R =1.66 0.33, RZZ =0.93 0.28 and R⌧⌧ =0.71 0.42. It is important to point out ± [281] D. L´opez-Val,± T. Plehn, M. Rauch, Measuring extended± Higgs sectors as a consistent free couplings that this talk was given in February, 2013, priormodel, to JHEP the 10 (2013) updates 134. arXiv:1308.1979 from Moriond., doi:10.1007/JHEP10(2013)134 We have. added a subsection regarding the Moriond updates below. In figure[282] S. 1 Kanemura, we present K. Tsumura, K.the Yagyu, points H. Yokoya, in Fingerprinting the (sin nonminimal↵, tan Higgs sectors,) plane Phys. that pass all the constraints in type I (left) and in type II (right)Rev. D90 (2014) at 1 075001. inarXiv:1406.3294 green and, doi:10.1103/PhysRevD.90.075001 at 2 in blue. Also. shown are the lines for the SM-like limit, that is sin( ↵) = 1 (negative sin ↵) and for the74 limit sin( + ↵) = 1 (positive sin ↵). To better understand these results we approximate RZZ in the case where Higgs production is due exclusively to gluon-gluon fusion via the top quark loop, and the total Higgs width is well approximated by (h b¯b). This is a very good approximation for most of the parameter space and it yields ! cos2 ↵ sin2 RI,approx = sin2 (↵ ) =sin2 (↵ ). (2) ZZ sin2 cos2 ↵
I,approx By setting RZZ = 1 one obtains the SM-like limit line shown in the picture for negative ↵. Applying the same simplified scenario we obtain for type II 1 RII,approx =sin2 (↵ ) . (3) ZZ tan2 ↵ tan2 Adding singlets
• Add only one scalar singlet, either real or complex • Transforms trivially under SM, i.e. is a singlet • Only couples to SM via the phi phi operator • Could be there is a whole hidden sector, SM blind to it 2 • V int = hs † S , for a real scalar In the quartic interaction, the self-interaction of S can get a non- • 2 zero vev, which is equal to the µ † term in SM. This mass scale is normally put by hand • They can give a natural candidate for dark matter, complex ones can give the right DM relic abundance
135 HiggsBounds-4.2.0 + HiggsSignals-1.3.0 ��� Parameters of real singlet 1 68.3% C.L. 2 ���� extension: ∆χ ���� 95.5% C.L. ��� v, tanbeta = v/vs, mh, mH, 20 99.7% C.L. alpha (mixing angle of hs 0.8 LEP excl. (95% C.L.) ��� hSM) LHC excl. (95% C.L.) -�� ���
15 [346] Z. Chacko, H.-S. Goh, R. Harnik, The Twin Higgs: Natural electroweak breaking from mirror sym-
] ��� metry, Phys. Rev. Lett. 96 (2006) 231802. arXiv:hep-ph/0506256, doi:10.1103/PhysRevLett. 0.6 96.231802.
| [347] B. Patt, F. Wilczek, Higgs-field portal into hidden sectorsarXiv:hep-ph/0605188. ��� α [ [348] R. M. Schabinger, J. D. Wells, A Minimal spontaneously- broken hidden sector and its impact 10 ��� on Higgs boson physics at the large hadron collider, Phys.� Rev. D72 (2005)���� 093007. arXiv:
�� hep-ph/0509209, doi:10.1103/PhysRevD.72.093007.
| sin [349] K. Assamagan, et al., The Higgs Portal and Cosmology, 2016. arXiv:1604.05324. � 0.4 URL http://inspirehep.net/record/1449094/files/arXiv:1604.05324.pdf Ratio of trilinear coupling to
��� [350] M. C. Bento, O. Bertolami, R. Rosenfeld, L. Teodoro, Selfinteracting dark matter and invisi- SM value, for vs = -75 GeV 5 bly decaying Higgs, Phys. Rev. D62 (2000) 041302. arXiv:astro-ph/0003350, doi:10.1103/ PhysRevD.62.041302. � Grey unphysical [351] C. P. Burgess, M. Pospelov, T. ter Veldhuis, The Minimal model of nonbaryonic dark mat- Pink, exclusion at 95% from ��� ter: A Singlet scalar, Nucl. Phys. B619 (2001) 709–728. arXiv:hep-ph/0011335, doi:10.1016/ 0.2 S0550-3213(01)00513-2. Δμ/μ�� Higgs signal 0 [352] V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf, G. Shaughnessy, Complex Singlet arXiv:0811.0393 doi:10. Pink lines, different values of Extension of the Standard Model,�� Phys.=- Rev.�� D79 ��� (2009) 015018. , ��� 1103/PhysRevD.79.015018. �� � ���� ���� the mixing angle sin^2 [353] G. Buchalla, O. Cata, A. Celis, C. Krause, Standard Model Extended by a Heavy Singlet: Linear 0 vs. Nonlinear EFTarXiv:1608.03564. Mhh (1,1) of the mass matrix ���[354] T. Robens,��� T. Stefaniak,��� Status of the��� Higgs Singlet���� Extension of���� the Standard Model���� af- 80 100 120 140 160 180 ter LHC Run 1, Eur. Phys. J. C75 (2015) 104. arXiv:1501.02234, doi:10.1140/epjc/ before diagonalization s10052-015-3323-y. �ϕ [���] m [GeV] [355] S. Ghosh, A. Kundu, S. Ray, Potential of a singlet scalar enhanced Standard Model, Phys. Rev. D93 (11) (2016) 115034. arXiv:1512.05786, doi:10.1103/PhysRevD.93.115034.
[356] D. Buttazzo, F. Sala, A. Tesi, Singlet-like Higgs bosons at present and future colliders, JHEP 11 (2015) 158. arXiv:1505.05488, doi:10.1007/JHEP11(2015)158. Figure 5: Exclusion regions for the real singlet extension. Right: Constraints on the[357] ( L.sin Bento, G.↵ C. Branco,,m GenerationH )-plane of a K-M phase from arising spontaneous CP breaking from at a high- the |energy scale, Phys.| Lett. B245 (1990) 599–604. doi:10.1016/0370-2693(90)90697-5136 . 2 [358] G. C. Branco, P. A. Parada, M. N. Rebelo, A Common origin for all CP violationsarXiv:hep-ph/ LEP and LHC exclusion limits as well as from the HiggsSignal distribution; reproduced0307119. with permission from [354].
[359] N. Darvishi, Baryogenesis of the Universe in cSMCS Model plus Iso-Doublet Vector Quark, JHEP Left: Exclusion at 95% C.L. from current Higgs couplings measurements at the LHC811 (2016) (light 065. arXiv:1608.02820 region),, doi:10.1007/JHEP11(2016)065 and deviation. in
2 [360] N. Darvishi, M. Krawczyk, CP violation in the Standard Model with a complex singletarXiv: the Higgs signal strengths µ/µSM sin ↵ (light solid lines). The ratio of the trilinear1603.00598 Higgs. coupling to its SM value ⌘ [361] R. Costa, M. M¨uhlleitner,M. O. P. Sampaio, R. Santos, Singlet Extensions of the Standard is drawn for vs = 75 GeV (dashed black lines). Dark gray: unphysical parameters.Model Reproduced at LHC Run 2: Benchmarks and with Comparison permission with the NMSSM, JHEP 06 (2016) from 034. arXiv:1512.05355, doi:10.1007/JHEP06(2016)034. from [356]. 79
Run 1 data. The black regions are still allowed. As shown by the vertical black stripe in Fig. 5, left, the sensitivity to sin ↵ is limited in the vicinity of the degenerate situation mh mH , where the alignment jump from the light to the heavy-mass scenario takes place. For the exact⇡ degeneracy the sensitivity vanishes since the mixing angle is undefined here. In the near-degenerate region, depending on the exact mass splitting, one can hope to get a better constraint from the high-resolution LHC channels, and ZZ⇤ 4`, or at the future colliders [271]. In the high mass region the strongest upper limit on sin ↵ arise from! the NLO corrections to the W mass and the perturbativity constraints, pushing this | | limit to below 0.2atmH =1TeV[354,366]. Relaxing the Z2-symmetry assumption in the real singlet model leads to minor complication of the analysis [356]. Fig. 5, right, reproduces the plot from [356] which shows, for a particular choice of 2 quartic coe cients, the values of µ/µSM sin ↵ and the triple Higgs coupling relative to the SM g /gSM as functions of the entry M 2 ⌘ 2 in Eq. (74) and the mass of the heavy scalar denoted hhh hhh hh ⌘ M as m . The dark region is unphysical, the light region is excluded by the LHC Run 1 Higgs data. 2 SM The white region is allowed and the lines show the predicted parameters sin ↵ and ghhh/ghhh.Future colliders will further constrain these two parameters. Citing [271, 356] for rough estimates, one expects 1 2 that the HL-LHC with 3 ab of data will probe sin ↵ down to 4–8% and yield the first indication ⇠ of the triple Higgs coupling. The future 100 TeV hadron collider should help detect deviations of ghhh + 2 from its SM value below 10%. The linear e e colliders aim at sin ↵ of about 2% level (ILC) or 0.3% level (CLIC). Using this plot, one can visualize the parameter range accessible to the future colliders. The case of a light second scalar, with mass in the GeV range, is severly constrained by the heavy meson decays and by non-observation of new long-lived particles which could produce displaced vertices [368, 369]. A moderately heavy scalar, but still lighter than 62.5 GeV, would be well visible in the decays of the SM-like Higgs boson. A review from 2008 of the possible collider signals in this regime can be found in [370]. Many of those opportunities are now of historic interest after the discovery of the 125 GeV Higgs bosons and placing an upper bounds on its invisible decay branching fraction. The collider phenomenology of the complex singlet extension depends on the scenario chosen. For two DM candidates, either mass-degenerate or not, the phenomenology is not dramatically di↵erent from the real singlet case: one expects missing ET events and, for light DM candidates, an extra contribution
45 HiggsBounds-4.2.0 + HiggsSignals-1.3.0 1 68.3% C.L. 2 95.5% C.L. ∆χ 99.7% C.L. 20 0.8 LEP excl. (95% C.L.) LHC excl. (95% C.L.) 15 0.6 | α 10
| sin 0.4 5 [346] Z. Chacko, H.-S. Goh, R. Harnik, The Twin Higgs: Natural electroweak breaking from mirror sym- metry, Phys. Rev. Lett. 96 (2006) 231802. arXiv:hep-ph/0506256, doi:10.1103/PhysRevLett. 0.2 96.231802. [347] B. Patt, F. Wilczek, Higgs-field portal into hidden sectorsarXiv:hep-ph/0605188.
[348] R. M. Schabinger, J. D. Wells, A Minimal spontaneously broken hidden sector and 0 its impact on Higgs boson physics at the large hadron collider, Phys. Rev. D72 (2005) 093007. arXiv: hep-ph/0509209, doi:10.1103/PhysRevD.72.093007. 0 [349] K. Assamagan, et al., The Higgs Portal and Cosmology, 2016. arXiv:1604.05324. 80 100 120 URL http://inspirehep.net/record/1449094/files/arXiv:1604.05324.pdf140 160 180 [350] M. C. Bento, O. Bertolami, R. Rosenfeld, L. Teodoro, Selfinteracting dark matter and invisi- bly decaying Higgs, Phys. Rev. D62 (2000) 041302. arXiv:astro-ph/0003350, doi:10.1103/ m [GeV]PhysRevD.62.041302.
[351] C. P. Burgess, M. Pospelov, T. ter Veldhuis, The Minimal model of nonbaryonic dark mat- Constraints on mH, |sinalpha| plane, andter: Afrom Singlet scalar,Higgs Nucl. Phys.signal B619 (2001) \Delta\chi^2 709–728. arXiv:hep-ph/0011335 , doi:10.1016/ Figure 5: Exclusion regionsmagenta for line the is realthe normalS0550-3213(01)00513-2 singlet Higgs extension.. Right: Constraints on the ( sin ↵ ,m )-plane arising from the [352] V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf, G. Shaughnessy, Complex Singlet H Extension of the Standard Model, Phys. Rev. D79 (2009) 015018. arXiv:0811.03932 , doi:10. | | LEP and LHC exclusion limits as well as1103/PhysRevD.79.015018 from the. HiggsSignal distribution; reproduced with permission from [354]. [353] G. Buchalla, O. Cata, A. Celis, C. Krause, Standard Model Extended by a Heavy Singlet: Linear Left: Exclusion at 95% C.L. from currentvs. Nonlinear Higgs EFTarXiv:1608.03564 couplings. measurements at the LHC8 (light region), and deviation in
[354] T. Robens,2 T. Stefaniak, Status of the Higgs Singlet Extension of the Standard Model af- the Higgs signal strengths µ/µSM tersin LHC Run↵ 1,(light Eur. Phys. J. solid C75 (2015) 104. lines).arXiv:1501.02234 The, doi:10.1140/epjc/ ratio of the trilinear Higgs coupling to its SM value 137⌘ s10052-015-3323-y. is drawn for vs = 75 GeV (dashed[355] blackS. Ghosh, A. lines). Kundu, S. Ray, Potential Dark of a singlet gray: scalar enhanced unphysical Standard Model, Phys. Rev.parameters. Reproduced with permission from from [356]. D93 (11) (2016) 115034. arXiv:1512.05786, doi:10.1103/PhysRevD.93.115034. [356] D. Buttazzo, F. Sala, A. Tesi, Singlet-like Higgs bosons at present and future colliders, JHEP 11 (2015) 158. arXiv:1505.05488, doi:10.1007/JHEP11(2015)158.
[357] L. Bento, G. C. Branco, Generation of a K-M phase from spontaneous CP breaking at a high- energy scale, Phys. Lett. B245 (1990) 599–604. doi:10.1016/0370-2693(90)90697-5.
[358] G. C. Branco, P. A. Parada, M. N. Rebelo, A Common origin for all CP violationsarXiv:hep-ph/ Run 1 data. The black regions are0307119 still. allowed. As shown by the vertical black stripe in Fig. 5, left, the [359] N. Darvishi, Baryogenesis of the Universe in cSMCS Model plus Iso-Doublet Vector Quark, JHEP sensitivity to sin ↵ is limited in the11 (2016) vicinity 065. arXiv:1608.02820 of, doi:10.1007/JHEP11(2016)065 the degenerate. situation mh mH , where the alignment [360] N. Darvishi, M. Krawczyk, CP violation in the Standard Model with a complex singletarXiv: 1603.00598. ⇡ jump from the light to the heavy-mass[361] R. Costa, M. M¨uhlleitner,M. scenario O. P. Sampaio, R. takes Santos, Singlet Extensions place. of the Standard For the exact degeneracy the sensitivity Model at LHC Run 2: Benchmarks and Comparison with the NMSSM, JHEP 06 (2016) 034. vanishes since the mixing anglearXiv:1512.05355 is undefined, doi:10.1007/JHEP06(2016)034 here.. In the near-degenerate region, depending on the exact mass splitting, one can hope to get a better79 constraint from the high-resolution LHC channels, and ZZ⇤ 4`, or at the future colliders [271]. In the high mass region the strongest upper limit on sin ↵ arise from! the NLO corrections to the W mass and the perturbativity constraints, pushing this | | limit to below 0.2atmH =1TeV[354,366]. Relaxing the Z2-symmetry assumption in the real singlet model leads to minor complication of the analysis [356]. Fig. 5, right, reproduces the plot from [356] which shows, for a particular choice of 2 quartic coe cients, the values of µ/µSM sin ↵ and the triple Higgs coupling relative to the SM g /gSM as functions of the entry M 2 ⌘ 2 in Eq. (74) and the mass of the heavy scalar denoted hhh hhh hh ⌘ M as m . The dark region is unphysical, the light region is excluded by the LHC Run 1 Higgs data. 2 SM The white region is allowed and the lines show the predicted parameters sin ↵ and ghhh/ghhh.Future colliders will further constrain these two parameters. Citing [271, 356] for rough estimates, one expects 1 2 that the HL-LHC with 3 ab of data will probe sin ↵ down to 4–8% and yield the first indication ⇠ of the triple Higgs coupling. The future 100 TeV hadron collider should help detect deviations of ghhh + 2 from its SM value below 10%. The linear e e colliders aim at sin ↵ of about 2% level (ILC) or 0.3% level (CLIC). Using this plot, one can visualize the parameter range accessible to the future colliders. The case of a light second scalar, with mass in the GeV range, is severly constrained by the heavy meson decays and by non-observation of new long-lived particles which could produce displaced vertices [368, 369]. A moderately heavy scalar, but still lighter than 62.5 GeV, would be well visible in the decays of the SM-like Higgs boson. A review from 2008 of the possible collider signals in this regime can be found in [370]. Many of those opportunities are now of historic interest after the discovery of the 125 GeV Higgs bosons and placing an upper bounds on its invisible decay branching fraction. The collider phenomenology of the complex singlet extension depends on the scenario chosen. For two DM candidates, either mass-degenerate or not, the phenomenology is not dramatically di↵erent from the real singlet case: one expects missing ET events and, for light DM candidates, an extra contribution
45 The direct searches are always more powerful than the Higgs signal strength measure- ments for low values of the resonance mass, while the opposite is true for high masses. While this is a general feature of direct and indirect searches, such an interplay was a h = h H = h priori not guaranteed in the125 physically interesting range of masses,125 and for the actual sen- Parameter Range Parameter Range sitivities of the future experiments. At the LHC, the point where the two searches become mh 124 – 128 GeV mh 3.5 – 124 GeV comparable varies from m 500 GeV for LHC8 to slightly more than 1 TeV at HL-LHC. mH 128φ – 1000 GeV mH 124 – 128 GeV + ∼ Also a linear e me−A collider,3.5 with – 40 GeV a sensitivity to deviationsmA in3.5 the – Higgs 40 GeV couplings in the
1% ballpark, wouldmH± become128 more – 1000 sensitive GeV than them HE-LHCH± upgrade128 – 1000 above GeV about 1 TeV (500 GeVThe for direct HL-LHC).tan searches are The0.5 always – same 50 more mass powerful of a TeV than sets thetan also Higgs the signal0.5 limit – strength 50 between measure- FCC-ee and
2 5 2 2 5 2 JHEP11(2015)158 FCC-hh.ments for lowm values12 of the10 – resonance 10 GeV mass, while the oppositem12 is10 true – 10 forGeV high masses. sin( ↵) 0.9 – 1 cos( ↵) 0.9 – 1 While this| is a general | feature of direct and indirect| searches, | such an interplay was a priori not guaranteed in the physically interesting range of masses, and for the actual sen- Table 1: 2HDM parameters and their ranges used for the scans. Left table for m = 125 GeV, 4sitivities Non minimal of the future supersymmetry experiments. At the LHC, the point where the two searches becomeh right for mH = 125 GeV. comparable varies from mφ 500 GeV for LHC8 to slightly more than 1 TeV at HL-LHC. + ∼ WeAlso can a now linear applye e− collider, the previous with a general sensitivity discussions to deviations to in some the Higgs concrete couplings models in the that are particularlythe1% Higgs ballpark, mass motivated alleviates would become scenarios the need more for sensitivefor large physics loop than contributionsbeyond the HE-LHC the SM. to upgrade achieve In thisabove its measured section about 1 we TeV value, consider thus thepossibly(500 Next-to-Minimal GeV allowing for HL-LHC). a more Supersymmetric natural The same sparticle mass spectrum Standard of a TeV sets [16 Model–22 also]. the (NMSSM), limit between which FCC-ee is the and minimal JHEP11(2015)158 deformationFCC-hh.The inclusion of the of MSSM a new singlet that includes scalar naturally a scalar also singlet leads in to the more spectrum. physical scalar A chiral particles: superfield one scalar and one pseudoscalar will be added giving in total three scalars (h1,2,3), two pseudoscalars S, singlet under the SM gauge groups, is coupled to the two Higgs doublets Hu and Hd of (a1,2), and the usual charged Higgs h±. A novel feature is that the discovered Higgs can be assigned the4 MSSM Non through minimal the supersymmetry superpotential to either h1 or h2. The latter possibility was found to be excluded in the MSSM by [23, 24]due to aWe combination can now apply of flavour the previous observables generalNMSSM and LHC discussions searches to for some scalars concrete decaying models to ⌧⌧ thatpairs, are though one might add that more recentlyNMSSM [25=] claimsMSSM there+ λ stillSH isuH ad very+ f( constrainedS) , possibility that(4.1) the particularly motivated scenariosW for physicsW beyond the SM. In this section we consider heavier scalar is the discovered one in the phenomenological MSSM. the Next-to-Minimal Supersymmetric Standard Model (NMSSM), which is the minimal whereThef( inclusionS) is a generic of the extra polynomial singlet superfield up to➝ third results order in a in modifiedS. superpotential, deformation• Add of the a MSSMsinglet that to includesthe MSSM a scalar Next singlet to in MSSM the spectrum. A chiral superfield This deformation has important consequences for the naturalness of the theory, given S, singlet under the SM gauge groups, is coupled to the two3 Higgs doublets Hu and Hd of The super potentialWNMSSM gets modified, SHuH whered + S f(S), is a polynomial of (2.2) the measured• value of the Higgs mass. Indeed, in the3 MSSM a 125 GeV Higgs requires the the MSSM throughat most the order superpotential three largest contribution to m to originate from SUSY-breaking effects, m2 m2 + ∆2, where where and are dimensionlessh coupling constants,b b b and web have assumed ah Z!3 invariantZ model. 2 = + λSH H + f(S) , (4.1) ∆Theis rest the of radiative the superpotential correctionWNMSSM is formed mainlyW fromMSSM due the to usual theu top-stop Yukawad terms sector for (see quarks e.g. and [61 leptons]). This as in in turnthe MSSM. requires Further, either one top needs squarks to add in the the multi-TeV corresponding range, soft supersymmetry or large trilinear breaking terms, terms both in cases the where f(S) is a generic polynomial up to third order in S. implyingscalar potential, a• largeThe tuning superpotential of the electroweak is scale. The ultimate reason for this is that the This deformation has important consequences for the naturalness of the theory, given supersymmetric couplingsNMSSM in the2 MSSM2 are controlled by the3 weak gauge couplings. The the measured valueV ofsoft the Higgsm mass.S S Indeed,+ A inHu theHdS MSSM+ A a 125S +h GeV.c. Higgs, requires the (2.3) situation is different in the NMSSM, | | where a supersymmetric3 contribution2 2 2 to the Higgs largest contribution to mh to originate from⇣ SUSY-breaking effects, mh !⌘ mZ + ∆ , where 2 masswhere∆ originatesmisS the, •A radiativeTheand at AHiggs tree-level correctionare mass dimensionful from is mainly now the mass due Yukawa-like to and the trilinear top-stop term parameters, sectorλSHu (seeHd, and e.g. one [61]). also This has in the other usualturn MSSM requires soft either SUSY top breaking squarks terms. in the multi-TeV range, or large trilinear terms, both cases 2 2 implyingAs the masses a large of tuning the singlet of the dominated electroweak2 λ v scalar scale.2 and The2 pseudoscalar ultimate2 2 reason are essentially for this is free that parameters, the mh ! s2β + mZ c2β + ∆ , (4.2) it openssupersymmetric the possibility couplings for them in theto be MSSM very2 light. are controlled If the singlet by the component weak gauge of a couplings.1 is large enough, The then suchsituation light particles is different can easily in the escape NMSSM, all whereexclusion a supersymmetric limits from earlier contribution searches. to We the briefly Higgs consider wherem as tan a functionβ vu/v ofd selectedis the ratio input of parameters, the vev’s of showing the two the Higgs results doublets in Fig. 1H.u,d Scan. As details manifest are a1mass originates≡ at tree-level from the Yukawa-like term λSHuHd, fromexplained (4.2), below. the most Relaxed natural constraints scenario have occurs been138 applied,when the apart coupling from thoseλ is of on order Higgs one, signal so rates. that Each horizontal bin is normalised suchλ2 thatv2 the largest bin in each row has contents = 1. This the largest contribution to mmh 2has a supersymmetrics2 + m2 c2 + ∆2 origin., We here briefly(4.2) sketch the h ! 2 2β Z 2β majorallows one features to see of which such value(s) a picture: of input parameter are preferred for a given ma1 . There are a few salient features to note. Most strikingly, panel (a) shows that A 0 or slightly negative is highly where tan β vu/vd is the ratio of the vev’s of the two Higgs doublets⇠ Hu,d. As manifest favoured for a≡ light a scenario. Panel (b) indicates some preference for 0.3, with another fromIt ( gives4.2), the a large most1 natural supersymmetric scenario occurs contribution when the coupling to theλ Higgsis of order mass. one, (with so that a different “hotspot”⋄ of points at 0.02 0.04. Panel (c) also shows a weak preference for a fairly small thedependence largest contribution on tan⇠β tothanm has the a D-terms) supersymmetric that is origin. sufficient We here to achieve briefly sketch 125 GeV the at tree- 0.15. h ⇠major features of such a picture: Whilstlevel. a scalar with mass 125 GeV is easily achievable in the NMSSM, it is useful to ⇠ momentarily review its dependence on the model input parameters. A scalar with mass 125 3 GeV It gives a large supersymmetric contribution to the Higgs mass (with a different± is achievable⋄ over the parameter range scanned. Fig. 2 shows the dependence of m on selected dependence on tan β than the D-terms) that is sufficient to achieve 125 GeV ath1 tree- level. – 19 –
–4– – 19 – • 7 physical Higgs bosons • Large SUSY contribution to the Higgs mass ➝ possible to achieve Mh = 125 GeV at tree level
• Allows larger masses for stops and gluinos O(λ/g)
• The coupling λ is not asymptotically free, λ җ 0.7 theory becomes strongly coupled, below the GUT scale
• λ ~ 1.2 strong coupling regime λ ~ 0.7 still perturbative at GUT scale
• If λ is sizeable ➝ lightest new particles are the extra scalar bosons of the Higgs sector
139 � λ ���
=Δμ μ�� JHEP11(2015)158 ϕ→/ �� �
Grey unphysical, pink and red [346] Z. Chacko, H.-S. Goh, R. Harnik, The Twin Higgs: Natural electroweak breaking from mirror sym- metry, Phys. Rev. Lett. 96 (2006) 231802. arXiv:hep-ph/0506256, doi:10.1103/PhysRevLett.
β ���� β excluded regions 96.231802. � ��� ��� Contours of fixed s2gamma [347] B. Patt, F. Wilczek, Higgs-field portal into��� hidden sectorsarXiv:hep-ph/0605188. [348] R. M. Schabinger, J. D. Wells, A Minimal spontaneously broken hidden sector and its impact Figure 8. NMSSM with “strong” coupling, λ =1.2 andon∆ Higgs= 70 boson GeV. physics Shaded at the regions large hadron and lines collider, as Phys. Rev. D72 (2005) 093007. arXiv: � JHEP11(2015)158 in figure 7. hep-ph/0509209, doi:10.1103/PhysRevD.72.093007���� .
[349] K. Assamagan, et al., The Higgs Portal���� and Cosmology, 2016. arXiv:1604.05324. URL http://inspirehep.net/record/1449094/files/arXiv:1604.05324.pdf 5 5 � Λ$0.7 [350] M. C. Bento, O. Bertolami, R. Rosenfeld,Λ$0.7 L. Teodoro, Selfinteracting dark matter and invisi- bly decaying Higgs, Phys. Rev.��� D62 (2000)��� 041302.���arXiv:astro-ph/0003350��� ����, doi:10.1103/ Μ#ΜSM Μ#ΜSM PhysRevD.62.041302. �ϕ ��� Φ&SM Φ&SM [ ] 4 [351]4 C. P. Burgess, M. Pospelov, T. ter Veldhuis, The Minimal model of nonbaryonic dark mat- ter: A Singlet scalar, Nucl. Phys. B619 (2001) 709–728. arXiv:hep-ph/0011335, doi:10.1016/ 0.01 Figure 8. NMSSM with “strong” coupling, λ =1.2 and ∆ = 70 GeV. Shaded regions and lines as S0550-3213(01)00513-2. in figure 7. [352] V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf, G. Shaughnessy, Complex Singlet Β 3 Β 3 Extension of the StandardProjected: Model, Phys. thin Rev. D79 solid (2009) 015018.red LHC13arXiv:0811.0393 , doi:10.
tan tan 1103/PhysRevD.79.015018. 5 thick solid red LHC14 5 [353] G. Buchalla, O. Cata, A. Celis, C. Krause, Standard Model ExtendedΛ$0.7 by a Heavy Singlet: Linear Λ$0.7 0.01 vs. Nonlinear EFTarXiv:1608.03564dashed. red 14 HL-LHCΜ#ΜSM Μ#ΜSM 2 2 [354] T. Robens, T. Stefaniak, Status of the Higgs Singlet ExtensionΦ& of theSM Standard Model af- Φ&SM ter LHC Run 1, Eur.4 Phys. J. C75 (2015) 104. arXiv:1501.02234, doi:10.1140/epjc/4 s10052-015-3323-y. 0.002 0.01 0.05 1 [355]1 S. Ghosh, A. Kundu, S. Ray, Potential of a singlet scalar enhanced Standard Model, Phys. Rev. D93 (11) (2016) 115034. arXiv:1512.05786, doi:10.1103/PhysRevD.93.115034. 200 400 600 800 1000 500 1000 Β 3 1500 2000 2500 Β 3
mΦ !GeV" [356] D. Buttazzo, F. Sala,mtan A.Φ ! Tesi,GeV Singlet-like" Higgs bosons at present and future colliders,tan JHEP 11 (2015) 158. arXiv:1505.05488, doi:10.1007/JHEP11(2015)158. 0.01 Figure 9. NMSSM with “perturbative” coupling, [357]λ =0L..7 Bento, and G.∆ C.= Branco, 80 GeV. Generation Shaded of regions a K-M phase and from spontaneous CP breaking at a high- energy scale, Phys. Lett.2 B245 (1990) 599–604. doi:10.1016/0370-2693(90)90697-5. 2 lines as in figure 7. [358] G. C. Branco, P. A. Parada, M. N. Rebelo, A Common origin for all CP violationsarXiv:hep-ph/ 0307119140. 0.002 0.05 [359] N. Darvishi, Baryogenesis1 of the Universe in cSMCS Model plus Iso-Doublet Vector Quark,1 JHEP 11 (2016) 065. arXiv:1608.02820, doi:10.1007/JHEP11(2016)065. are no particular differences with respect to the analysis of section 2, given200 that400vs, λ and600 800 1000 500 1000 1500 2000 2500 !GeV" !GeV" λS are free parameters also in the NMSSM. However,[360] N. Darvishi, as we M. have Krawczyk, already CP violation discussed, in themΦ Standard the Model with a complex singletarXiv: mΦ 1603.00598. SM NMSSM prefers low values of Mhh. Accordingly, the deviations from one in ghhh/ghhh are expected to be milder than in the generic case[361] (seeR. figures Costa, M.1 and M¨uhlleitner,M.Figure2), but 9 stillO.. NMSSM P. allowing Sampaio, with R. for Santos, “perturbative” Singlet Extensions coupling, of theλ Standard=0.7 and ∆ = 80 GeV. Shaded regions and Model at LHC Runlines 2: Benchmarks as in figure and7 Comparison. with the NMSSM, JHEP 06 (2016) 034. observable effects at the LHC. arXiv:1512.05355, doi:10.1007/JHEP06(2016)034.
79
are no particular differences with respect to the analysis of section 2, given that vs, λ and – 22 – λS are free parameters also in the NMSSM. However, as we have already discussed, the SM NMSSM prefers low values of Mhh. Accordingly, the deviations from one in ghhh/ghhh are expected to be milder than in the generic case (see figures 1 and 2), but still allowing for observable effects at the LHC.
– 22 – Dark Matter?
Many models provide DM-DM interactions • candidates for DM • • Most models assume sterile neutrinos, WIMP • neutrinos scalars, Higgses, K-K • Usually require a • scalars, others discrete symmetry to make them stable • axion (solution to the strong CP problem), • Constraints from pseudoscalar cosmology: relic density, • SUSY particles, structure formation, neutralino, sneutrino, BBN axino Decaying DM • DM could be more than • one type of particle
141 in certain cases is can be additionally enhanced [410]. Its absence at tree level is essentially due to orthogonality of charged Higgses to the Goldstone modes, which is most clearly seen in the Higgs basis. Such a vertex would require that three factors v, W ± and Z come from the first doublet with vev, and one field H⌥ comes from extra doublets without vev, which is impossible to organize. In GM model,
we have additional charged Higgses in the 5-plet H5±, which reside inside and are not orthogonal to the Goldstone modes. ScalarAnother firm prediction dark of multi-doublet matter models is that popular the HWW and HZZ couplingsnow for each neutral Higgs are less than one (52) and that their squares sum up to unity. This sum rule is required by the unitarization of the longitudinally polarized vector boson scattering at high energies, in particular + + 2 2 W W W W , in which the Higgs exchange cancels the E /v growth of the cross section [221]. In triplet! models, this sum rule is modified because the doubly charged scalar can now propagate in the u-channelNatural and compensate in models for the overshooting with by extended the HV V couplings Higgs [411]. sectors • Finally, we mention that the idea behind the GM model is not specific to triplets and can be extended toand even higher global multiplets. symmetries, One should however belike careful Z2 not to run into troubles with perturbative unitarity. Logan and Rentala [412] explored all such generalized GM models not violating these con- straints, the largest one being GCM6 with three complex sextets. Their general scalar potentials were analyzed and an overview of phenomenological features was presented. One strong conclusion was an These symmetries may remain unbroken afterh eW • absolute upper bound on the SM-like Higgs coupling to VV possible in the GM framework: V < 2.36. symmetry breaking DM candidate stable 6. Astroparticle and cosmological implications⟹ 6.1. Scalar dark matter models
D SM SM D
D D D SM SM D
D SM SM SM SM D SM D D SM D (semi-)annihilation direct detection production
Figure 7: Complementary ways to explore models with DM candidates. Left: annihilation or semiannihilation processes, Building and testing modelseither in with early extended Universe Higgs or at sectors present, middle: DM scattering in DM direct detection experiments, right: pair production Igor P. Ivanov,of arXiv:1702.03776 new states possibly [hep-ph]. with a cascade decay inside the dark sector. 142 Scalar dark matter (DM) candidates naturally arise in extended Higgs sectors equipped with global symmetries, under which the SM fields remain invariant while the extra scalars transform non-trivially. In a large part of the parameter space these symmetries stay intact after EWSB and protect the lightest non-trivially transforming scalar against decay into SM fields. An attractive feature of such models is that presence of DM candidates can manifest itself in three complementary channels: annihilation, scattering, and production, Fig. 7, which can be probed via very di↵erent observables. Let us denote the DM candidates by D and the heavier scalars from the same multiplet by D0. Then annihilation DD SM, coannihilation DD0 SM, or semi-annihilation DD D+SM processes a↵ect evolution of the! early hot Universe and determine! the DM relic abundance.! They also lead to present day signals such as -rays emission from galactic centers. Scattering processes may produce signals in DM
50 • annihilation, co-annihilation and semi-annihilation will determine DM abundance
• also missing energy signature in colliders • Example: minimal dark matter. Add extra scalars in higher dimensional reps of SU(2), only renormalizable interactions in Lagrangian.
• If lightest component neutral ⟹ DM candidate • They are used also to explain small neutrino masses
143 Real singlet extension
• One real gauge singlet and a Z2 symmetry • Symmetry makes DM stable and prevents it from mixing with neutral part of the doublet • Correct DM relic abundance for some values of the coupling λ between SM-DM • Cannot be too heavy (above TeV), or runs into trouble with perturbative constraints
144 ���
Fermi
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6.1.2. Inert doublet model 2HDM can easily incorporate scalar DM candidates. Taking Type I 2HDM and postulating the exact Z2 symmetry (43) in the scalar potential, one can obtain 2 =0inalargepartoftheparameter h i 52 Inert doublet model space• without2HDM any fine-tuning type 1 with [188]. Z2 The symmetry scalars from thehas second, natural “inert” DM doublet are odd under Z2 and are protectedcandidate against decay into the pure SM final states. Due to their mass splitting, the inert scalars can sequentially decay into one another but the lightest scalar h1 is stable and serves as the DM candidate. Pairwise interactions between these scalars are still possible, leading to DM annihilation and co-annihilation•
= processes 0, i.e. inert, as well as in to large collider parts signatures. of parameter space, This modelodd known under as theZ2 Inert doublet model (IDM) was resurrected a decade ago [434, 435, 436] and its phenomenology has been explored in great detail. It is easily doable with analytical calculations, it can beHeavier implemented scalars in computer from codes, this anddoublet its parameter decay space into is the relatively lightest small, allowing for e cient• parameter scans. It is predictive and can be strongly constrained by the present and future data. Severalone comprehensive analyses of astroparticle and collider constraints on the IDM parameter space have been published [437, 438, 439, 440, 441]; below we borrow some illustrative examples from [441]. DM annihilation, co-annihilation and collider signatures • 2 The scalarpossible potential of IDM has the form (42) without the m12, 6,and 7 terms. All of its free parameters are real and satisfy the usual stability bounds (44). In addition, one assumes 5 4 > 0 which guarantees that the vacuum is neutral and that the charged inert scalars are heavier| | than the lightest• neutralMasses one. are The scalar spectrum in the inert sector is
2 1 2 2 2 2 1 2 2 M + = v + m ,M,M = ( + )v + m . (83) h 2 3 22 h1 h2 2 3 4 ⌥ | 5| 22
In this way, 5 4 defines the splitting between the charged and the lightest neutral scalars, while alone shows| | the splitting between the two neutrals h and h . The notation (H, A)isoftenused | 5| 1 2 instead of (h1,h2)anditalludestothespecific146CP-parity assignment of the two neutral scalars. As we explained in section 3.4, this assignment is a matter of convention. Although h1 and h2 indeed have opposite CP-parities, which is manifest via the Zh1h2 vertex, it is impossible to unambiguously assign which of them is CP-even and which is CP-odd. In the absence of direct coupling to fermions, the model has two CP-symmetries, h1 h1,h2 h2 and h1 h1,h2 h2, which get interchanged upon the basis change i . Either! can be! used as “the ! CP-symmetry”! of the model, making the 2 ! 2 specification of the CP properties of h1 and h2 abasisdependentstatement.
The five IDM free parameters can be chosen, for example, as the three masses Mh1 , Mh2 >Mh1 , and M + , the combination of quartic couplings + , which determines the SM-DM h 345 ⌘ 3 4 | 5| interaction vertex h125h1h1, and the parameter 2 > 0, which governs the self-interaction inside the inert sector. Stability and perturbative unitarity places theoretical bounds on 345 and 2. The values of 345 strongly a↵ect phenomenology, while 2 remains essentially unconstrained by data. To illustrate how the collider data and astrophysical observations constrain the parameter space of 2 IDM, we show in Fig. 9 the results for the DM relic density ⌦DMh [441] obtained with the MicrOMEGAs 2.4.1 package [138]. The left and right plots correspond, respectively, to quasi-degenerate h1,h2 and + + + h masses, Mh2 = Mh = Mh1 +1 GeV, and large mass splitting Mh2 = Mh = Mh1 +100 GeV; on both plots, several scenarios with positive and negative 345 are shown. The relic density result obtained by Planck 2 Planck ⌦DM h =0.1184 0.0012 [427] is shown by the horizontal line. If one leaves room for other mechanisms which can produce± additional DM candidates, this value should be considered as an upper bound rather than the absolute value to be fitted within IDM. Several features are visible on these plots. For the light DM masses at small mass splitting, the curves go below the Planck value and display two dips at 40 and 45 GeV, corresponding to the resonant coannihilation h h Z and h h+ W +. These processes are governed by the gauge couplings, and 1 2 ! 1 ! they e ciently reduce the relic density even at zero 345. Since the inverse processes, the W and Z boson decays to light inert scalars, would also take place, this region is already closed by the LEP data.
+ For large mass splitting, the dips disappear since Mh1 + Mh2 or Mh1 + Mh are above MZ and MW , respectively. Coannihilation is suppressed, and the relic density can be brought below the Planck value only at large 345 > 0.3. However, these large values are already ruled out by the non-observation of | | ⇠ 53 Figure 9: The DM relic density as a function of Mh1 for various 345 parameters for small (left) and large (right) mass splitting. The horizontal line corresponds to the relic density obtained by Planck [427]. Reproduced from [441]. quasi degenerate and large mass splitting limits in IDM
small mass region[439] ruledA. Ilnicka, out M. Krawczyk, by LHC T. Robens, and Inert Planck Doublet Model in light of LHC Run I and astrophys- the invisible Higgs decay at the LHC: the upperical limit data, Phys. of Br(Rev. D93h125 (5) (2016)inv) 055026. IDM can also produce signals for direct [448][454] andE. Nezri, indirect M. H. G. Tytgat, DM G. Vertongen, search e+ experiments and anti-p from inert doubletvia heavy model dark inert matter, scalar annihilation, which can be detectable viaJCAP -rays 0904 (2009) [449, 014. arXiv:0901.2556 450, 451], ordoi:10.1088/1475-7516/2009/04/014 via its neutrino [452,. 453] and cosmic-ray signals [454]. Collider signatures of IDM include mono-Z85 or mono-jet + missing ET signatures [438, 439, 441], dijets or dijets plus dileptons with missing ET [455, 456], multilepton signals [457, 458, 456, 440], modified rates of the SM-like Higgs decays to and Z [459, 460, 444, 461], long-lived charged scalars and other exotic signals. Prospects of detecting and exploring IDM at ILC 54 Flavour/family symmetries more symmetries • Add the flavour • Discrete symmetry of your ZN, A4, S3, S4, Q6, choice Δ27... → might also generate • Continuous → upon accidental continuous breaking might generate symmetries massless Goldstone bosons U(1), SU(3)... • Explain masses and mixings of quarks and leptons • Usually also add more particles → Higgs 148 Flavour symmetries • Mass hierarchy of quarks and charged leptons very hierarchical • Mass hierarchy also in neutrino sector, but whether normal or inverted hierarchy unknown • Small mixing in quarks, but large mixing in leptons • Perhaps there is an underlying flavour/family symmetry 149 • Abelian or non- • Discrete or Abelian? continuous? • Abelian symmetries • Breaking of have all uni- continuous dimensional irreps symmetries leads to Goldstone bosons • Need non-Abelian symmetries, have • Discrete symmetries irreps of more than have several one dimension ⟹ representations of possible to explain small dimension the patterns of ⟹ appropriate to masses and mixings describe three generations 150 • Non-Abelian, discrete groups widely studied recently • Permutational symmetries: SN, A N • Dihedral symmetries: DN • Double valued dihedral symmetries: DN’ Other double valued groups: T’, O’ • Subgroups of SU(3): Δ(3n2), Δ(6n2) 151 3HDM with S3 • Low-energy model • 3HDM without symmetry: 57 couplings in the Higgs • Extend the concept of potential flavour to the Higgs sector by adding two more eW doublets • Add symmetry: permutation symmetry of three objects, symmetry operations (reflections and rotations) that leave an equilateral triangle invariant 152 Why a particular symmetry? • Prior to the eW symmetry breaking all families look the same → permutation symmetry • Smallest non- Abelian group Logarithmic plot of fermions discrete group S3 • Has irreducible representations: 2, 1S, 1 A Mass ratios 153 Predictions, advantages? • Possible to reparametrize • Underlying symmetry in quark, mixing matrices in terms of leptons and Higgs → residual mass ratios, successfuly symmetry of a more fundamental one? • Reproduces well CKM → one less parameter as SM • Lots of Higgses: 3 neutral, 4 charged, • PMNS → fix one mixing angle, 2 pseudoscalars predictions for the other two Natural decoupling limit within experimental range • Further predictions will come • Predicts reactor angle �13 ≠ 0 from Higgs sector: decays, branching ratios • No extra flavons • Higgs potential has 8 couplings A. Mondragón, M. M., F. González, E. Peinado, U. Saldaña, O. 154 Félix, E. Rodríguez, A. Pérez, H. Reyes... A4 symmetry • Symmetry group of a tetrahedron, even permutations of four objects • Symmetry has to be broken Observations softly to reproduce the • Four irreducible mixing pattern Massesrepresentations: of the charged fermions 1, 1’, 1’’, are 3 strongly hierarchical 8 4 Deviations4 2 from tri- mu : mc : mt λ : λ :1,md : ms : m•b λ : λ :1, ≈ ≈ Fermions can be nicely5 2 bimaximal mixing may come • me : mµ : mτ λ : λ :1 where λ θC 0.22 assigned to the≈ 3 ≈ from≈ higher dimensional Mass hierarchy in the ν sector is milder, ordering tilloperators now unknown. from neutrino • It predicts the tri-bimaximal sector Mixing parameters:mixing, which is now ruled small mixingsout by for quarks, large mixings for leptons. for leptonexperiment mixing special structures are allowed: “tri-bimaximal” (TB): (1 σ) 2 TB 1 2 TB 1 2 TB sin (θ12 )= 3 , sin (θ23 )= 2 , sin (θ13 )=0. “µ-τ”symmetric(MTS): 2 MTS 1 2 MTS sin (θ23 )= 2 , sin (θ13 )=0. 155 All these issues need a theoretical description: Flavor sym metry GF ! ⇒ LAUNCH, MPI-K, Heidelberg – p. Add a discrete flavour symmetry to a SUSY GUT theory Interplay between the different symmetries (and their breaking) would lead to an explanation of the fermionmass hierarchies 156 • SU(5)xA4, SU(5)xQ6, SU(5)xS4xU(1), SU(5)xT7… • SO(10)xS3, SO(10)xS4, SO(10)xD4, SO(10)xA4, SO(10)xΔ(54)… • Many of these models also have extra Abelian discrete symmetries ZN, besides the GUT and the flavour symmetries 157 • The addition of discrete symmetries might lead to accidental continuous symmetries • After eW symmetry breaking there might be surprises ⟹ unwanted Goldstone bosons from the continuous symmetries • There might also be residual discrete symmetries ⟹ can spoil the pattern of masses and mixings 158 Extra U(1)’s • Well motivated, they appear naturally when breaking large unification groups to SM • Also appear in string compactifications • In SUSY GUTs and string compactifications U(1)’ and SU(2)xU(1) breaking tied to soft SUSY breaking • At TeV, proposed to alleviate the quadratic divergences in the Higgs mass • Dynamical symmetry breaking models often include U(1)’s • In some models associated with almost-hidden sectors 159 How do we sieve through models/ideas • Interplay between • Applies to theory and theory and experiment experiment • Models have to • Theoretical provide predictions uncertainties have to and/or explanations to be taken into account the SM open questions (not only experimental) • Each extension BSM has assumptions that • Most successful have to be spelled out models combine more explicitly to be able to than one idea test it 160 Going BSM — assumptions • Usually do not break Lorentz or gauge invariance These symmetries are too fundamental to mess around with… • Respect experimental results (not everybody likes to) These are facts of life, even if for some of them origin not understood • Guiding principles: mathematical consistency and experimental compatibility • Many assumptions are a matter of “taste”, acknowledge it 161 Where (in energy) are these models realized? • Depends... • some models are valid around few TeV and a “bit” above • some models assumed valid to TeV to the GUT or Planck scale • some models valid only at GUT or Planck scale • If they are described by a QFT we can use Renormalization Group Equations (RGE) to test them at different scales 162 String theory GUTS �R Extra SUSY other Higgses symmetries 163 Conclusions • There is unknown physics to explore → BSM • To discover it we need an interplay between mathematical consistency (theories), experiments and observations in particle physics, cosmology and astrophysics • There is a lot of work to do... 164