Supersymmetry. Theory, Experiment, and Cosmology

The Minimal SuperSymmetric Model (MSSM) 99

Note that, in the case where the scales associated with supersymmetry are large (MW M2, |µ|), one of the charginos is predominantly wino whereas the other one is mostly Higgsino (see Exercise 5).

Similarly, the neutralino mass term reads (with obvious notations, λB , resp. λ3, is the gaugino associated with the U (1)Y gauge field Bµ, resp. the SU (2)L gauge field

A3µ)

λBL

1

λ3L

λ

λ

Ψc

Ψc

Ln = − 2

H20 R

Mn

0

+ h.c.

(5.49)

B R 3R

H10 R

ΨH2 L

ΨH10 L

with

M1

0

MZ cos β sin θW

−MZ sin β sin θW

Mn =

0

M2

−

MZ cos β cos θW

MZ sin β cos θW

MZ cos β sin θW

−

MZ cos β cos θW

0

µ

−

M

Z

sin β sin θ

W

+M

Z

sin β cos θ

W

µ

0

−

−

(5.50)

where M1 and M2 are, respectively, the U (1)Y and SU (2)L soft supersymmetry breaking gaugino mass terms. The eigenstates are written χ01, χ02, χ03, χ04 in increasing mass order (we will often denote the lightest neutralino simply by χ0 or χ and other neutralinos by χ ). More precisely, one writes

χ01L

χ02L

= Nχ03L

χ04L

λBL

λ

3L

,ΨH10 L

ΨH20 L

χ01Rχ0

2R = Nχ03R

χ04R

λBRλ3R

ΨcH10 R

Ψc 0

H2 R

, (5.51)

where N is a unitary matrix (the second equation derives from charge conjugation: λ = λc) which satisfies:

N MnN −1 = diag mχ10 , mχ20 , mχ30 , mχ40	.	(5.52)

By construction, the neutralinos are Majorana spinors.

We note that, in the limit MZ M1 < M2 < |µ|, the lightest neutralino is mostly bino (λB ), the next to lightest mostly wino (λ3) and the heaviest are Higgsinos (ΨH10 ± ΨH20 ) (see Exercise 5).

We see that, at tree level, the -ino sector depends on two supersymmetric parameters (µ, tan β) and three soft supersymmetry breaking mass terms (M1, M2, M3). We will see in the next chapter that the assumption of gaugino mass universality at the gauge unification scale yields relations among the latter three parameters.

100 The minimal supersymmetric model

5.3.3The squark and slepton sector

Assuming that the squark mass matrices can be diagonalized (in family space indexed by i {1, 2, 3}) simultaneously with those of the corresponding quarks, we have

	m2	m2	!
Mu2i =	uiLL	uiLR
	m2	m2
,
	uiRL	uiRR

=

,2 =

Mdi

=

m2Qi + m2ui + 16 (4MW2 − MZ2 ) cos 2β

mui (Aui − µ cot β)

!

m2diLL m2diLR m2diRL m2diRR

m2Qi + m2di − 16 (2MW2 + MZ2 ) cos 2β

mdi (Adi − µ tan β)

mui (Aui − µ cot β) !

,

m2Ui + m2ui + 23 (−MW2 + MZ2 ) cos 2β (5.53)

mdi (Adi − µ tan β) !

,

m2Di + m2di + 13 (MW2 − MZ2 ) cos 2β (5.54)

where mui = λui v2, mdi = −λdi v1 are the fermion masses. Also m2Qi , m2Ui and m2Di are the soft masses, respectively, for the SU (2)L scalar doublet Qi and singlets Ui and

Di, and Aui , Adi are the trilinear soft terms (defined as in (5.8)):

= m2

u

u + d d

m2 u

u

m2

d

d

Lsoft −

Qi

iL iL −

−

Di

iR

iR

iL

iL

Ui

iR

iR

−

·

·

(5.55)

Adi λdi

qiL

H1 diR + Aui

λui qiL

H2 uiR + h.c. .

We will identify in turn the origin of each term. This will provide a good illustration

of the di erent interactions obtained earlier. For simplicity, we consider only up-type squarks. One finds in the entries of the mass-squared matrix (5.53):

•Supersymmetric mass terms arising from the superpotential after gauge symmetry breaking. For example, the terms

−µH1 · H2 + λuQ · H2U c

in

the

superpotential

(5.2)

yield

a term

|

λ

Q

·

H

|

2 in

|

dW/dU c

2 and

−

1

2

2

2

u

2

|

1

λ

u0

1

U U c

µH0

in dW/dH0

. After gauge symmetry breaking ( H0

= v ,

H2

=

v2), this

yields

quadratic

terms in the scalar potential, respectively,

λ2 v2u˜

L

u˜

= m

2 u˜

L

u˜ and, arising

from the cross-product in the second case,

u 2

L

u

L

−µ λuv1u˜L u˜R + h.c. = −µ mu cot β + h.c.

The Minimal SuperSymmetric Model (MSSM) 101

•Supersymmetric mass terms arising from the D-terms after gauge symmetry breaking. The D-term reads

	1																	2				1									i							2
																											g
VD =				g						φ iti3j φj +																						φ iyiφi! + · · ·
		2					i,j																2					2
=	1 g2 1 u˜ u˜										L	+		1 H		0 H0										1 H					0 H0 +			· · ·			2
	2					2			L					2	1						1		− 2						2			2
	+			1	2				1	u˜ u˜					1	H				0	H			0		+			1		0	H	0	+					2	.	(5.56)
				2	g				6				L	−	2									1					2		H		2		· · ·
										L								1													2
After symmetry breaking ( H10 = v1, H20 = v2), the cross-terms yield:
							1									1										1
									(v12 − v22)										g2 −								g 2				u˜L u˜L ,
							2										2									6

or more generally

12 (v12 − v22) g2t3i − g 2yi/2 φ iφi,

which can be expressed in terms of MZ , MW and β using (5.23) and cos 2β =

(v12 − v22)/(v12 + v22):

qiMW2 − (yi/2)MZ2 cos 2β φ iφi.

•Soft supersymmetry-breaking scalar mass terms, respectively, m2Q and m2U for u˜L and u˜R .

•Mass terms arising from the soft-supersymmetry-breaking A-terms in (5.55) after gauge symmetry breaking.

Similarly for charged leptons, one has

,				2		2
M	2	=	meiLL		meiLR
				2		2
	ei		meiLR meiRR
		=	mL2 i + me2i + 21 (MZ2 − 2MW2 ) cos 2β
					mei (Aei − µ tan β)

!

mei (Aei − µ tan β)

.

m2Ei + m2ei + (MW2 − MZ2 ) cos 2β

(5.57)

We note that, in all three cases (u, d, e-type sfermions), the nondiagonal terms are proportional to the corresponding fermion mass. They are nonnegligible only in the case of the third family. Thus, for f = t, b or τ , the mass eigenstates are

˜	=	cos θf	sin θf	˜
f1				fL
f˜2		− sin θf	cos θf	= f˜R	,	(5.58)
with corresponding eigenvalues m˜ f2		1 < m˜ f22 and 0 < θf < π.

102 The minimal supersymmetric model

5.4Baryon and lepton number

Realizing supersymmetry by doubling the number of fundamental fields has the immediate consequence of losing a remarkable property of the Standard Model: renormalizability and SU (3) × SU (2) × U (1) gauge symmetry impose automatic baryon (B) and lepton (L) number conservation. Indeed, it is possible to generalize baryon and lepton number to the supersymmetric particles (squarks, sleptons) but the introduction of these new fields allows us to write new renormalizable couplings which are B or L violating. These couplings were written in (5.4) and, if all are present, they lead to rapid proton decay.

In order to prevent this, one may choose to impose B or L global symmetries. This is not a favored solution of the problem because global symmetries, if not protected by a local symmetry, tend to be broken by gravitational interactions. One prefers to obtain them as a consequence of a broken local symmetry.

In Section 5.2.2, we have discussed the introduction of R-parity which may arise from the breaking of a continuous R-symmetry. Such a R-parity allows us to set to zero all renormalizable interactions breaking B or L number (cf. (5.4)) since Q, L, U c, Dc and Ec have R-parity −1 whereas H1 and H2 have R-parity +1.

The question must however also be addressed at the level of nonrenormalizable interactions. Since a supersymmetric generalization of the Standard Model usually arises as a low energy e ective theory of a more fundamental theory, nonrenormalizable terms are expected, with coe cients of the order of the mass scale M of the underlying theory to some power. This power is fixed by the dimension of the corresponding term. We will see later explicit examples in the context of grand unification or string theory.

A thorough discussion therefore requires to list all the possible terms according to their dimensions. Since we are working in a supersymmetric set up, we write them as F -terms or D-terms. We recall a few facts from Chapter 3:

•an F -term of a product of superfields with total dimension n is a term in the Lagrangian with dimension n + 1;

•a D-term of a product of superfields and possibly their hermitian conjugates, with total dimension n, is a term in the Lagrangian with dimension n + 2.

Then, imposing R-parity leaves us the possibility of writing the following gauge invariant terms at dimension 5:

(LLH2H2)	F	, (QQU cDc)		, (QU cLEc)	F	, (QQQL)	F	, (U cU cDcEc)	F	,
			F
LEcH2† D,			QDcH2† D,	QU cH1† D,

where SU (2) and SU (3) indices are contracted in order for the corresponding term to be singlet under these groups. Each term has a coupling of order M −1. It is already clear at this stage that R-parity is not su cient to prevent baryon nor lepton violation: the terms (QQQL)F and (U cU cDcEc)F violate both. It is thus important that either the fundamental scale M be superheavy (as in grand unification) or extra symmetries be imposed.

5.5The LSP and dark matter

We have seen above that models with R-parity include a stable particle: the lightest supersymmetric particle (LSP). We first show why this particle is an excellent

The LSP and dark matter 103

candidate to account for cold dark matter. We then discuss its nature as well as its properties.

5.5.1The LSP as a WIMP

We will not recall here the arguments that lead to the need for dark matter. It su ces to say that the total matter content ρM of the energy density in the Universe is believed to be

ΩM ≡	ρM	0.3,	(5.59)
	ρc

where ρc = 3H02m2P 10−26kg/m3 is the critical density (corresponding to a spatially flat spacetime)9. On the other hand, the density of baryonic matter ρB is limited to be

ΩB ≡	ρB	≤ 0.02.	(5.60)
	ρc

This leaves a lot missing, under the form of nonbaryonic matter or of a more exotic component.

Let us see under which conditions a given particle of mass mX might provide the right amount of dark matter. We will suppose that this particle is neutral and colorless, otherwise it would have some observable e ects through scattering on matter. There are two competing e ects to modify the abundance of this species: annihilation and expansion of the Universe. Indeed, the faster is the dilution associated with the expansion, the least e ective is the annihilation because the particles recede from one another. This is summarized in the following Boltzmann equation which gives the evolution with time of the particle number density nX :

				dnX	2	(eq)2
					+ 3HnX = −σannv nX	− nX		,	(5.61)
				dt
where		σ			¯ annihilation cross-section times the
			annv is the thermal average of the XX			(eq)	is	the	equilibrium density,
relative		velocity of the two particles annihilating,				nX

and H the Hubble parameter. When the temperature drops below the mass mX , the annihilation rate becomes smaller than the expansion rate and there is a freezing of the number of particles in a covolume. We study this freezing for a general species in Section D.3.3 of Appendix D, to which we refer the reader. We find in the case of cold relics (i.e. relics which are nonrelativistic at the time of freezing) that the freezing temperature Tf is given in terms of the variable xf = mX /(kTf ) by equation (D.73) of this appendix:

	MP mX σannv			gX	! ,
xf = ln 0.038						(5.62)
	h¯2			g1/2

where gX is the number of internal degrees of freedom of the particle and g is the total number of relativistic degrees of freedom present in the Universe at the time of decoupling. One checks that xf is a quantity of order one (more precisely 20).

9See Appendix D for a more detailed treatment.

104 The minimal supersymmetric model

The relic density of particles is given by

mX nX (t0)

s0 h¯3

g1/2

Ω

≡

= 40

π

xf

(5.63)

X

5 k H02MP3

gs

σannv

ρc

where s0 is the present entropy density and gs an e ective number of degrees of freedom which enters in the expression of the entropy (see equation (D.64) of Appendix D). Putting the explicit values, one finds

Ω		h2		1.07 × 109 GeV−1		xf	.	(5.64)
				g1/2MP
	X	0				σannv

We note for further reference that the smaller the annihilation cross-section is, the larger is the relic density. We find ΩX (100 TeV)−2( σannv )−1 10−13 barn/ σannv (in units where h/ = c = 1, 1 barn = 2.5 × 103 GeV−2).

Thus ΩX will be of order 1 (more precisely a fraction of 1) if σannv is of the order of a picobarn to a femtobarn, which is a typical size for an electroweak process. Also writing dimensionally

σannv	α2	,	(5.65)
	mX2

where α is a generic coupling strength, we find that ΩX is of order 1 for a mass mX α × 100 TeV, i.e. in the TeV range. This is why one is searching for a weakly interacting massive particle (WIMP).

As we have seen earlier, in supersymmetric models with R-parity, the lightest supersymmetric particle (LSP) is stable and provides a good candidate: supersymmetric particles are massive (as it results from negative searches) and many of them are weakly interacting (gauginos, Higgsinos and sleptons). In the following section, we will discuss the possible nature of the LSP.

The simple analysis that we presented is somewhat more involved when the WIMP appears within a system of particles with which it can annihilate, as is the case in the supersymmetric set up that we study: such coannihilations are particularly helpful to decrease the relic density in the case where the WIMP is almost degenerate with some of the other particles [40, 207].

To illustrate this, let us consider a system of two Majorana particles X1 and X2

of masses m1 < m2. Their abundance is determined by the following reactions:
annihilation				, i = 1, 2		(5.66)
	XiXj → ff
scattering	X2f → X1f,					(5.67)
decay						(5.68)
	X2 → X1ff ,

where f and f stand for light Standard Model particles (we have assumed the conservation of a X-parity). The Boltzmann equations describing the evolution

The LSP and dark matter 105

with time of their respective

number densities n1

and

n2 read [40] (compare

with (5.61)):

dn1

(eq)2

(eq)

(eq)

+ 3Hn1 =

f &− σann11

v n12

− n1

− σann12 v n1n2 − n1

n2

dt

+ σscat2→1v n2nf(eq) − σscat1→2v n1nf(eq) + Γ21

n2 − n2(eq)'

(5.69)

dn2

(eq)2

(eq)

(eq)

+ 3Hn2 =

f &− σann22

v n22

− n2

− σann12 v n1n2 − n1

n2

dt

− σscat2→1v n2nf(eq) + σscat1→2v n1nf(eq) − Γ21

n2 − n2(eq)'

(5.70)

, σscati→j

= σ (Xif → Xj f ) and Γij =

with obvious notation: σannij = σ XiXj → ff

X

≡

1

2

,

Γ X

i → Xj ff . Since all the X2

which survive annihilation eventually decay into X

1

the relic density will be the present value of n

n + n . Summing the previous two

equations, we obtain the evolution equation for nX :

dnX

ij

(eq)

(eq)

+ 3HnX = − i,j

σannv

ninj − ni

nj

.

(5.71)

dt

This analysis can straightforwardly be generalized to a system of N particles Xi (i =

1, . . . , N ) with gi internal degrees of freedom and masses mi (m1 ≤ m2 ≤ · · · ≤ mN ).

#N

Equation (5.71) applies to nX ≡ i=1 ni and allows us to determine the relic density of the only stable particle of this system, i.e. X1.

To simplify (5.71), one notes [207] that the reaction rates are much faster for (5.67) and (5.68) than for (5.66): this ensures that the relative number densities of each species remain at their equilibrium value: ni/nX n(eq)i /n(eq)X . Then, (5.71) simply reads, just as (5.61),

dnX

e

2

(eq)2

+ 3HnX

= − σannv nX

− nX

,

dt

where we have defined

ni(eq) nj(eq)

e

N

ij

σann ≡

σann

,

i,j=1

nX(eq)

nX(eq)

with (see (D.60) of Appendix D)

ni(eq)

= gi mi 3/2 e(1−mi/m1)x1 ,

ge

= N gi

mi

3/2 e(1−mi/m1)x1 ,

nX(eq)

ge

m1

i=1

m1

where x1 = m1/(kT ).

(5.72)

(5.73)

(5.74)

106 The minimal supersymmetric model

The decoupling temperature Tf and relic density ΩX1 are now given by

m1

M

P

m1 σe v

ge

! ,

x1f ≡

= ln

0.038

ann

kTf

h¯2

g1/2

Ω

h2

1.07 × 109 GeV−1

x1f

.

(5.75)

X1

0

1/2

MP

σe v

g

ann

We see that, in the case of near degeneracy in mass, and equal degrees of free-

dom and cross-sections, ge N g1 and ΩX1 /ΩX = x1f /xf 1 + (ln N )/xf 1 + (ln N )/20, where xf and ΩX are the quantities computed ignoring coannihila-

tions (i.e. using (5.62) and (5.63)). A more important e ect arises if the coannihilators Xi (i ≥ 2) have a larger annihilation cross-section: this increases σanne v with respect to σannv and may decrease ΩX1 with respect to ΩX by several orders of magnitude.

Another possible deviation from the simple analysis presented earlier arises when one approaches poles in the annihilation cross-section [207]. Indeed, if the exchange of a particle of mass mA and width ΓA is possible in the s-channel (in supersymmetric models, this often turns out to be the pseudoscalar A, hence the notation), the estimate

(5.65) is more correctly written
σannv	4α2s	,	(5.76)
	(s − mA2 )2 + ΓA2 mA2

where s = 4m2X in the limit of zero velocity. We see that, close to the pole i.e. for m2A 4m2X , the cross-section reaches the value σannv|pole = 4α2/Γ2A. If the resonance is narrow, this might lead to a significant increase in the cross-section and decrease in the relic density.

5.5.2Nature of the LSP

If the LSP had nonzero electric charge or color, it would have condensed with baryonic matter and should be present today in anomalous heavy isotopes. The absence of such isotopes puts very stringent bounds on such particles, which therefore could not play the rˆole of dark matter.

A possible candidate would be a sneutrino. Direct searches (see Figure 5.3) exclude a LSP sneutrino in a mass window between 25 GeV and 5 TeV [28]. But lighter sneutrinos have been excluded by searches at LEP and, on the other side of the allowed spectrum, a mass larger than several TeV represents a rather unnatural value for a LSP. In the context of the theories that we consider here, this discards the sneutrino as a LSP.

We are thus left10 with the lightest neutralino χ01 ≡ χ which, as we have seen in Section 5.3.2, is a combination of gauginos and Higgsinos.

10Another possibility is that the lightest supersymmetric particle is the supersymmetric partner of the graviton, the gravitino (see Chapter 6).

The LSP and dark matter 107

We list below for further use the main interactions of the lightest neutralino (for a more complete treatment, see [242]). First, the coupling to the neutral vectors is given

by:

1

g

|N13|2 − |N14

|2 χγ¯ µγ5χ Zµ,

LχχZ = −

(5.77)

4

cos θW

or more generally

0

=

1 g

Z

χγ¯

µLχ0

(N

N

−

N

N

)

−

χγ¯

µRχ0

(N

N

−

N

N

) ,

4 cos θW

LχχmZ

µ

m

13

m3

14

m4

m

13

m3

14

m4

(5.78)

where the N matrix is defined in (5.51), m = 1, . . . , 4, and L, R = (1 γ5)/2. We note that such interactions probe the Higgsino content of the neutralino LSP. The coupling to charginos and W reads:

1

Lχχr± W = −gχγ¯ µ N12ZLr1 −

√

N14ZLr2 L

2

+ N12ZRr1 +

1

N13ZRr2

χr+Wµ− + h.c.

(5.79)

√

R

2

where the ZL, ZR matrices are defined in (5.45).

Next, the couplings to the neutral Higgs system can be summarized in

χχH =

g

(N12

tan θ

N11)

(cos αN13

sin αN14)H0

(sin αN13 + cos αN14)h0

L

2

−

W

0

−

−

(5.80)

+i(sin βN13 − cos βN14)A

χRχ¯ + h.c.

We see that such couplings vanish when the lightest neutralino is purely gaugino (N13 = N14 = 0) or purely Higgsino (N11 = N12 = 0). Moreover, if the N matrix elements are real, the coupling of χ to h0 and H0 (resp. A0) is purely scalar (resp. pseudoscalar).

Finally, the couplings to quarks and squarks are given by

where

Lχqq˜ = −q¯iL χ Xiq˜iL + Ziq q˜iR − q¯iR χ (Yi q˜iR + Ziqq˜iL ) + h.c.

(5.81)

= −g√

ti3N12 + (yi/2) tan θW N11 ,

Xi

2

√

Yi

= g

2 qi N11 tan θW ,

(5.82)

u

g

mu

N

d

g md

N

=

√

i

14

=

√

i

13

Zi

,

Zi

,

MW sin β

MW cos β

2

2

where i = 1, 2, 3 is a family index (we have neglected quark and squark mixings).

5.5.3LSP annihilation

We assume from now on that the LSP is the lightest neutralino χ. We first give in Table 5.4 the main annihilation channels for a neutralino at rest. We note that the

108 The minimal supersymmetric model

Table 5.4 Main annihilation channels of the lightest neutralino χ at rest, with the indication of the type of exchanged particle (and the corresponding channel). The couplings λAf f are given in Table 5.3.

Final state Exchanged		Amplitude
s	t, u

¯	˜
f f	f
¯	A
f f
¯	Z
f f
W +W −	χr±
ZZ	χ0
	m

g2 1 +

mχ

−

mχ

mf˜

2

mf

2

−1

gλAf f

(N12 − tan θW N11)(sin βN13 − cos βN14)

4 − mA2 /mχ2 + iΓAmA/mχ2

g2

mf mχ

|

N13

|

2

− |

N14

|

2

cos2 θW

MZ2

g2 1

2

−

2

2

2

2

2

2

χ

1

χ

W

χ

1

2

N12Z

N14Z

+

N12ZRr1 +

N13ZRr2

MW

Lr1

√2

Lr2

−

√2

χr±

− m

1 + m

/m

M

/m

2

g2

1 −

MZ2

N13Nm3 − N14Nm4

cos2 θ

m2

1 + m2 0 /m2

−

M 2

/m2

W

χ

χm

χ

Z

χ

amplitudes for most channels behave dominantly as m2χ/MSUSY where MSUSY is a supersymmetric mass, except for the Z exchange (which dominates when it is open). The Higgsino fraction N13 or N14 plays a significant rˆole: it enhances the annihilation cross-section and thus tends to decrease the relic density for a Higgsino-like LSP.

¯

Finally, the annihilation amplitude into f f is proportional to the fermion mass and thus important only for the third family.

Other annihilation channels at rest include W ±H (through the exchange of A in the s-channel or charginos in the t, u channels), ZH or AH and Zh or Ah (through the exchange of Z or A in the s-channel or neutralinos in the t, u channels). They turn out to be less relevant in explicit computations, except when one considers light scalars. We also note that decays to photons or gluons arise only at the one loop level.

As is clear from the considerations of Section 5.5.1, the neutralino χ is not at rest at the time of freezing. Typically, its average velocity is given by v2 = 6kT /m = 6/x. One may then expand the averaged cross-section as σv = a + b v2 + · · · . The term a is the S wave contribution already discussed, whereas the term b contains contributions from S and P waves. New contributions appear when one considers the decay of a neutralino LSP which is not at rest (for example s-channel exchange of Z0, h0 and

H0 for the decay into W +W − pairs), as well as new final states: Z0A0, A0A0, h0h0,

H0H0, H0h0 and H+H−.