Electroweak and Higgs boson internal bremsstrahlung. General considerations for Majorana dark matter annihilation and application to MSSM neutralinos
Received: May
Electroweak and Higgs boson internal bremsstrahlung. General considerations for Majorana dark matter annihilation and application to MSSM neutralinos
Torsten Bringmann 1 3
Francesca Calore 0 1
Ahmad Galea 1 3
Mathias Garny 1 2
0 Universite Savoie Mont Blanc , 27 Rue Marcoz, 73000, Chambery , France
1 9 Chemin de Bellevue , BP110, AnnecyleVieux, 74941, Annecy Cedex , France
2 Technical University Munich , JamesFranckStr. 1, D85748 Garching , Germany
3 Department of Physics, University of Oslo , Box 1048, NO0371 Oslo , Norway
It is well known that the annihilation of Majorana dark matter into fermions is helicity suppressed. Here, we point out that the underlying mechanism is a subtle combination of two distinct e ects, and we present a comprehensive analysis of how the suppression can be partially or fully lifted by the internal bremsstrahlung of an additional boson in the nal state. As a concrete illustration, we compute analytically the full amplitudes and annihilation rates of supersymmetric neutralinos to nation of two standard model fermions, plus one electroweak gauge boson or one of the ve physical Higgs bosons that appear in the minimal supersymmetric standard model.
Beyond Standard Model; Cosmology of Theories beyond the SM; Supersym
1 Introduction 2 Majorana dark matter and relevant symmetries
Chiral symmetry, gauge symmetry and helicity suppression
Lifting of Yukawa and isospin suppression
Gauge invariance in IB processes
3
Neutralino annihilation to f f and an additional nalstate particle
4
5
2.1
2.2
2.3
3.1
3.2
3.3
3.4
3.5
4.1
4.2
5.1
5.2
5.3
Full analytic amplitudes and gaugeinvariant subsets
Helicity suppression in the MSSM 3.2.1 3.2.2
Yukawa suppression
Isospin suppression
Yukawa and isospin suppression lifting
Heavy propagator suppression
Threshold e ects
Narrow width approximation and total cross section
Spectrum of stable particles
Double counting issues
Results for the MSSM
Theoretical benchmark models
Total annihilation rate
Yield enhancement
5.4 Indirect detection spectra
6
Conclusions A Neutralino annihilation amplitudes
A.1 Expansion of amplitudes in the helicity basis
A.2 Results for expanded amplitudes
A.3 Suppression lifting from individual diagrams
B
Numerical implementation C Spin correlations of decaying resonances
{ 1 {
Introduction
The prime hypothesis for the cosmologically observed dark matter (DM) [1] is a new type of
elementary particle [2]. Among theoretically wellmotivated candidates, weakly interacting
massive particles (WIMPs) play a prominent role.
This is because such WIMPs very
often appear in theories that attempt to cure the netuning problems in the Higgs sector
of the standard model of particle physics (SM), and because thermal relics with weak
masses and cross sections at the electroweak scale are typically produced with the correct
abundance to account for the DM density today [
3, 4
]. Another advantage is that the
WIMP hypothesis can be tested in multiple ways: at colliders, where the signature consists
in missing energy, in direct detection experiments aiming to observe DM particles recoiling
o the nuclei of deep underground detectors, and in indirect searches for the debris of
DM annihilation in cosmic regions with large DM densities. Direct detection experiments
have become extremely competitive in constraining smaller and smaller scattering rates [5,
6], and collider searches have pushed the scale of new physics to TeV energies in many
popular models [
7, 8
]. It is worth stressing, however, that only `indirect' searches would
eventually allow to test the WIMP DM hypothesis in situ, i.e. in places that are relevant for
the cosmological evidence for DM. Also indirect searches have become highly competitive
during the last decade, now probing the `thermal cross section' (the one that is needed to
produce the observed DM abundance) up to WIMP masses of the order of 100 GeV [9, 10].
A key quantity for both thermal production of WIMPs and indirect searches is the
total annihilation cross section. Multiplied by the relative velocity v of the incoming DM
particles, it can in the non relativistic limit be expanded as
v = a + bv2 + O(v4) :
(1.1)
It was noted early [11, 12] that radiative corrections to v can be huge because of
symmetries of the annihilating DM pair in the v ! 0 limit. For indirect DM searches, changes in
either the partial cross section, for a given annihilation channel, or the di erential cross
section, d v=dE, may be phenomenologically even more important. The reason is that
an additional photon in the
nal state can give rise to pronounced spectral features in
the DM signal in both gamma [13] and charged cosmic rays [14]. For electroweak
corrections, the situation is in some sense even more interesting because, on top of the just
mentioned e ects, completely new indirect detection channels may open up. In this way,
antiproton data can for example e ciently constrain DM annihilation to light leptons when
considering the associated emission of W or Z bosons [15]. In the presence of pointlike
interactions, such as described by e ective operators, the resulting spectra can be computed
in a modelindependent way by using splitting functions inspired by a parton picture [16].
This approach is very useful for generic DM phenomenology and is, for example, the one
implemented in the `cookbook' for indirect detection [17]. One of the main results of this
article (see also [18]) is that the resulting cosmic ray spectra from DM annihilation can
di er substantially from the actual spectra, calculated in a fully consistent way from the
underlying particle framework.
{ 2 {
Here we revisit in detail one of the most often discussed examples where radiative
corrections can be large, namely the case of a Majorana DM particle . The treelevel
annihilation rate into light fermions f is then on general grounds `helicity suppressed', for
v ! 0, as a consequence of the conserved quantum numbers of the initial state [19]. The
resulting suppression by a factor of mf2 =m2 can be lifted by allowing for an additional
vector [11] or scalar [20] boson in the nal state, implying that for DM masses at the
electroweak scale the radiative `corrections' can be several orders of magnitude larger than the
result from lowest order in perturbation theory.1 Here, we revisit these arguments and point
out that the e ect commonly referred to as helicity suppression is in fact the culmination
of two distinct suppression mechanisms, in the sense that they can be lifted independently.
This results, in general, in a rather rich phenomenology of such radiative corrections.
As an application, we consider electroweak corrections to the annihilation cross
section of the lightest supersymmetric neutralino  one of the most often discussed DM
prototypes [28] and still a leading candidate despite null searches for supersymmetry at
ever higher energies and luminosities at the LHC [
7, 8
]  though our main
ndings can
be extended in an analogous way to other DM candidates that couple to the SM via the
electroweak or Higgs sector. Concretely, we provide a comprehensive analysis, both
analytically and numerically, of all threebody
nal states from neutralino annihilation that
contain a fermion pair and either an electroweak gauge boson or one of the ve Higgs bosons
contained in the minimal supersymmetric standard model (MSSM), for a neutralino that
can be an arbitrary admixture of Wino, Bino and Higgsino.2
We nd large parameter
regions where these threebody
nal states signi cantly enhance the DM annihilation rate,
with the impact on the shape of the cosmicray spectra relevant for indirect detection being
even more signi cant.
One of the technically most involved aspects, apart from the shear number of diagrams
to be considered, is how to avoid `double counting' the onshell parts of the threebody
amplitudes that are already, implicitly, included in the corresponding twobody results.
We provide an indepth treatment of this issue and demonstrate how to accurately treat
not only the total cross section but also the resulting cosmicray spectra. We again
nd
signi cant e ects on the latter, indicating the need to correctly adopt this method also
for other DM candidates. In fact, in order to reliably test the underlying particle models,
our ndings suggest that at least for fermionic
nal states it is in general not su cient
to use the modelindependent spectra traditionally provided by numerical packages. The
numerical routines that implement our results for the neutralino case will be fully available
with the next public release of DarkSUSY [39, 40].
1The lifting of helicity suppression via threebody nal states is also relevant for real scalar dark
matter [21{25] and, under certain conditions, for vector dark matter [26]. The case in which the additional
boson is a Z0 has been considered in [27].
2For neutralino annihilation, so far only the cases of photon [13] and gluon [29] internal bremsstrahlung
for pure binos in [30{35], for Higgsinos in [36], and for pure Winos in [37]. A
rst study for a general
neutralino has been performed in [18]. Finally,
nal states involving the SMlike Higgs boson have been
considered in [20] for a toy model encompassing a pure Bino (see also [38]).
{ 3 {
This article is organized as follows. We start in section 2 with a general discussion of
Majorana DM annihilating to fermions and the relevant symmetries that arise for v ! 0,
revisiting in particular the often invoked `helicity suppression' arguments and how this
suppression can be lifted fully or partially. In section 3 we then consider the concrete
case of neutralino DM, and discuss the various possibilities of how the presence of an
additional nal state boson can add sizeable contributions to, or even signi cantly enhance,
the twobody annihilation rates. The double counting issues mentioned above are then
addressed in detail in a separate section 4. We scan the parameter space of several MSSM
versions and demonstrate the e ect of these newly implemented corrections to neutralino
annihilation in section 5, both for the annihilation rates and the cosmicray spectra relevant
for indirect DM searches, and present our conclusions in section 6. In a more technical
appendix, we describe the details of our analytical calculations to obtain the threebody
matrix elements for fully general neutralino annihilation in the MSSM (appendix A), the
numerical implementation of these results in DarkSUSY (appendix B), and how to correctly
treat spin correlations of decaying resonances (appendix C).
2
Majorana dark matter and relevant symmetries
For DM annihilation in the Milky Way halo, where DM particles have typical velocities of
order 10 3, only the rst term in eq. (1.1) gives a sizeable contribution. In the following
we therefore neglect p and higher partial wave contributions, and it is understood that
all (di erential) cross sections are e ectively evaluated in the zerovelocity limit. For an
swave, the relative angular momentum in the initial state is L = 0. Due to the Majorana
nature the initial particles are identical, but because we consider fermions the total wave
function still needs to be antisymmetric with respect to exchanging the incoming particles.
The orbital wave function for L = 0 is symmetric, so in order to get an antisymmetric total
wave function the spins must couple into an antisymmetric state. This is only possible for
the singlet state, with S = 0, resulting in the following quantum numbers:
J = 0;
C = ( 1)L+S = 1;
P = ( 1)L+1 =
1 :
(2.1)
Here, the general expressions for C and P apply because we have a system of two fermions.
Assuming no signi cant sources of CP violation in the theory, which generally are highly
constrained by measurements of the electric dipole moment and other precision
experiments, the symmetry of the
nal state is hence also restricted to be JCP = 0 . This
implies the wellknown `helicity' suppression of the annihilation rate into light fermions,
similar to the case of charged pion decay. In the following we rst brie y review the origin
of this suppression, and then argue that it can in fact be related to a combination of two
rather independent suppression mechanisms.
2.1
Chiral symmetry, gauge symmetry and helicity suppression
We want to study Dirac fermions f as possible nal states from the annihilation of Majorana
mf )f , which is invariant under
Lorentz transformations, i.e. invariant under SU(2)L+R. In the massless limit, mf ! 0,
{ 4 {
where the arrows indicate the spin direction along the zaxis (the rst entry refers to the
HJEP09(217)4
antifermion, the second to the fermion). If the two fermion momenta are (anti)parallel 
e.g. because they are emitted backtoback as the nal states of a DM annihilation process
 the zaxis can be chosen to be aligned in the same direction as the momenta, and the
above spin projections on the z axis are directly related to the helicities of the two particles.
Choosing pf (pf ) to point along the positive (negative) zaxis, the helicity con gurations
h = Szpz= jpzj of the singlet and triplet state are then given by
(2.2)
(2.3)
(2.4)
(2.5)
and
1
p
2
+ ;
1
2
1
2
;
1
2
1
2
1
; p
2
1
2
1
2
+ ; +
1
2
;
1
2
1
2
1
2
1
2
+ + ; +
;
1
2
! p
fR; fL
fL; fR
1
2
; +
;
1
2
this symmetry is upgraded to a chiral symmetry SU(2)L
SU(2)R, in which the left and
right handed Weyl states transform independently of each other, and helicity and chiral
eigenstates unify. For a fermion pair f f , the spins can combine to either a singlet (S = 0),
or a triplet (S = 1) spin state,
(j"#i
j#"i) =p2;
n
j##i ; (j"#i + j#"i) =p2;
o
j""i ;
!
fL; fL ; p
fR; fL + fL; fR
; fR; fR
where the arrows indicate the chiral states in the left/right decoupling limit, i.e. for mf ! 0.
The momentum con guration thus restricts which helicity states can be associated to
the spin states. Angular momentum and the assumed CP invariance, on the other hand,
restrict which spin state can be realized. Since CP = ( 1)L+S
( 1)L+1 = ( 1)S+1, for
example, only the singlet state with S = 0 is compatible with the odd CP parity of the
initial state. Eq. (2.4) then tells us that both fermion and antifermion in this momentum
con guration must have the same helicity. In any chirally symmetric theory, however,
the antifermion must necessarily have the opposite helicity of the fermion. We note that
angular momentum conservation alone leads to the same conclusion: since L = r
p, we
must have Lz = 0 and hence Sz = Jz = 0. Eqs. (2.4), (2.5) then imply that fermion and
antifermion must, independently of the value of S, have the same helicity. The annihillation
process
! f f is therefore only possible if chiral symmetry is broken in the Lagrangian,
for example through an explicit fermionic mass term mf fLfR or through the coupling of
the fermion f to a scalar eld
fLfR. It follows that the amplitude of the annihilation
process must be proportional to the chiral symmetry breaking parameters mf or .
In addition, it is instructive to consider also the isospin of the involved particles. Since
leftand righthanded SM fermions transform under di erent representations of SU(2)L, the
nal states fLfR and fRfL have total isospin I = 1=2. The initial state
, on the other
{ 5 {
hand, has necessarily integer isospin, implying
I 6= 0. The annihilation rate thus has to
vanish for an unbroken SU(2)L, and therefore has to be proportional to at least one power
of the Higgs vacuum expectation value (VEV) vEW . For heavy DM the ratio v
vEW =m
becomes small, and processes with
I 6= 0 will be suppressed by some power of v.
In total, this implies that the amplitude for
! f f has to involve (at least) one
parameter that breaks chiral symmetry, and one power of vEW that controls breaking
of the SU(2)L symmetry. For the SM fermions that receive their mass from the Higgs
mechanism, both of these conditions are ful lled for the usual helicity suppression factor
mf . Depending on the model, however, there can be further possibilities, as we will discuss
in detail for the case of the MSSM below, and it is useful to discriminate between the two
suppression mechanisms. In the following, we therefore refer to the suppression related to
chiral symmetry breaking as Yukawa suppression, and to the one related to electroweak
symmetry breaking as isospin suppression. While isospin suppression is controlled by only
one parameter, v = vEW =m , there can in principle be several sources of chiral symmetry
breaking, for example in models with more complicated Higgs sectors. Nevertheless, as
we discuss in detail in section 3, all terms that break chiral invariance in the MSSM are
accompanied by Yukawa couplings yf / mf =vEW . Even though the following discussion
of suppression lifting is completely model independent we will thus continue to assume, for
concreteness, that chiral symmetry breaking is controlled by yf .
2.2
Lifting of Yukawa and isospin suppression
With the above discussion in mind, the only way to avoid the suppression of nonrelativistic
Majorana DM annihilation is to allow for an additional nal state particle. Lorentz
invariance requires this additional particle to be a boson, such that the leading process we are
interested in is of the form
! BF f
(2.6)
where B is a scalar or vector boson, and F = f if B is electrically neutral. The additional
boson can be either a SM particle, in particular a photon (refered to as electromagnetic
IB), a gluon, a weak gauge boson (W , Z) or the Higgs boson h, or it can be a new particle
beyond the SM (for example a heavy Higgs boson within the MSSM). For the moment we
want to keep the discussion modelindependent, and therefore focus on the former case. A
frequently used approximation is to restrict the discussion to B being radiated o a fermion
line in the nal state, as described by soft and/or collinear splitting functions [16, 17]. We
emphasize that this approach does not capture the (partial) lifting of helicity suppression,
and therefore is inadequate for the case of heavy Majorana DM annihilation to fermions.
Taking the gauge restoration limit vEW ! 0, it becomes straightforward to exhibit the
scaling of a given process with yf and vEW . (We emphasize that we consider this limit only
in order to discuss the possible mechanisms of Yukawa and isospin suppression lifting, while
all our numerical results later on take the full dependence on vEW and yf into account). In
this limit, the left and righthanded components of the fermions in the nal state and in
internal lines can not only be considered as gauge interaction eigenstates but as independently
propagating degrees of freedom. The fermion mass is treated perturbatively in the mass
in{ 6 {
FRfL or FLfR
nal state boson B and fermion combinations. Entries / 1 correspond to
processes that potentially can lift both Yukawa and isospin suppression of the twobody process.
Entries / yf can lift isospin suppression but are still suppressed by the Yukawa coupling, while
those / vEW can lift Yukawa suppression but are still suppressed by v = vEW =m for large m .
sertion approximation, and is associated with a chirality ip along with a suppression factor
mf / yf vEW . In addition, longitudinally polarized gauge bosons WL=ZL can be replaced
by the corresponding Goldstone bosons G ; G0 by virtue of the Goldstone boson
equivalence theorem, cf. eqs. (2.7){(2.8) below. All nal states thus have de nite SU(2)L quantum
numbers (i.e. I = 1=2 for G ; G0; h; fL, I = 0 for fR, and I = 1 for WT ), except for the ZT ,
which is a mixture of I = 0; 1 states (even in the gauge restauration limit, we nd it
convenient to express our results in terms of the Z boson instead of the neutral SU(2)L boson).
The amplitude of the generic threebody process indicated in eq. (2.6) can be nonzero
for vEW
! 0 only if
I = 0, i.e. if isospin is conserved. Furthermore, the amplitude
must vanish for yf ! 0 unless both fermions have the same chirality. Note that this is
possible for threebody processes because the kinematics does not force the fermions to
be emitted backtoback in the centerofmass (CMS) frame, and therefore the arguments
discussed in section 2.1 do not apply.3
These two observations immediately determine
which annihilation processes can lift either Yukawa or isospin suppression (or both). In
these considerations, for various combinations of fermion chiralities and
nal state bosons
(where the longitudinal gauge bosons represent the corresponding Goldstone bosons). Both
suppression factors can be lifted only in processes where a transverse gauge boson (ZT ,
WT , , or gluon g) is emitted and the
nal state fermions are described by spinors of
equal chirality (FRfR or FLfL). For longitudinal gauge bosons (ZL or WL) or the Higgs
boson h, only one of the suppression factors can (potentially) be avoided for threebody
nal states: isospin suppression can be lifted if the fermions are of opposite chirality, and
Yukawa suppression can be lifted if the fermions are of equal chirality.
3In the extreme case where both fermions are emitted in the same direction, e.g., one simply has to
exchange fR $ fL in eqs. (2.4), (2.5), which allows equal chiralities of the fermions in both the singlet and
triplet spin state. In this kinematical con guration, it is easy to visualize how the fermion momentum can be
balanced by the emitted boson B, and how their spin can combine with SB and L to the required J = 0 for
both SB = 0 and SB = 1. In general, the spin singlet and triplet states will be linear combinations of all
chiral states, with expectation values that depend on the angle between the fermion momenta, thus rendering
the above argument essentially independent of the speci c kinematical con guration. Also the requirement
of CP conservation is much less restrictive for threebody than for twobody
nal states. A general
discussion is somewhat complicated by the fact that e.g. F f is not necessarily a CP eigenstate that could be
analysed individually, but in principle straightforward by classifying all possible e ective operators that
Let us stress that the symmetry arguments presented above simply guarantee that the
amplitude must vanish for yf ! 0 and vEW ! 0, respectively, and the same applies to any
gauge invariant subsets of diagrams. The actual suppression can thus be stronger than
indicated by table 1, i.e. by additional powers of vEW or yf . At the same time, we caution
that single diagrams can scale in a di erent way, depending on the gauge choice, such that
the vanishing for yf ! 0 or vEW ! 0 is in general not guaranteed.
Following up on the last comment, let us for convenience brie y recall how to verify gauge
independence and identify gauge invariant subsets of diagrams. While for photon emission
a good test is to check whether a given set of diagrams satis es the Ward identity M (
!
f f )k
= 0, where k is the momentum of the photon, this does not work for electroweak
IB because SU(2)L
U(1)Y has been spontaneously broken. Indeed the question of gauge
invariance changes in general, as weak hypercharge and isospin are no longer conserved
in their original form. For the spontaneously broken GlashowWeinbergSalam theory the
correct way to de ne gauge invariance is in terms of the preserved BRST symmetry [42, 43],
under which SM
eld transformations involve ghost elds which arise from the electroweak
gauge xing procedure. This implies a new set of Ward identities, which in general depend
on the choice of gauge. Using the standard R class of gauges [44], we arrive at the Ward
identities for electroweak IB as expected from the Goldstone equivalence theorem:
M (
M (
! f f Z)k
! F f W )k
= imZ M(
= mW M(
! f f G0) ;
! F f G ) :
(2.7)
(2.8)
We reiterate that eqs. (2.7) and (2.8) in general apply to (subsets of) the full amplitude,
not individual diagrams, and are a valuable test for the results outlined in the next section.
3
Neutralino annihilation to f f and an additional
nalstate particle
In this section we apply the general discussion of helicity suppression lifting in Majorana
DM annihilation to the lightest supersymmetric neutralino as DM candidate, and additional
nal state bosons charged under SU(2)L. For photon or gluon IB we refer to the references
listed in the introduction. Concerning the choice of DM candidate, we note that much of
the following discussion is still rather generic and can thus be extended in a straightforward
way to any theory with an extended Higgs sector or where the DM particles belong to a
di erent electroweak multiplet. We will introduce the relevant threebody processes and
Feynman diagrams in section 3.1, revisit the discussion of the helicity suppression in light
of the speci c situation encountered in the MSSM (section 3.2) and then demonstrate in
detail how these suppressions can be lifted, fully or partially, in section 3.3. In sections 3.4
and 3.5,
nally, we discuss two mechanisms by which threebody cross sections can be
enhanced which are not related to the helicity suppression of twobody nal states.
{ 8 {
F¯
V/S
f
V/S
F¯
F¯
f
F¯
V/S
V/S
χ˜01
χ˜01
χ˜01
f˜i
f˜i
V/S
f
F¯
f
F¯
V/S
χ˜01
χ˜01
χ˜01
χ˜01 f˜i
χ˜0n/χ˜±n
χ˜0n/χ˜±n
F¯
V/S
f
f
F¯
χ˜01
χ˜01
). See text for more details on how the individual topologies
are referred to in this article.
HJEP09(217)4
3.1
Full analytic amplitudes and gaugeinvariant subsets
From now on, we thus assume DM to be composed of the lightest neutralino,
which is a superposition of Wino, Bino and Higgsino states,
f
F¯
f
V/S
F¯
V/S
~01,
(3.1)
(3.2)
(3.3)
obtained by diagonalizing the neutralino mass matrix
= N11B~ + N12W~ 3 + N13H~10 + N14H~20 ;
M1
0
0
B
B
B
B
B
p
2
2
0
M2
2
pgv2
2
pg0v1 gv1
p
Higgsino mass parameter; v1 and v2 are the VEVs of the two Higgs doublets, with vEW =
v1=v2, and g and g0 are the SU(2)L and U(1)Y couplings, respectively.
We follow the conventions of ref. [45], as implemented in DarkSUSY, and take all mass
eigenvalues to be positive, while the diagonalization matrix N can be complex.
We want to consider here all threebody
nal states that contains a fermion pair and
a boson that is charged under SU(2)L. Assuming CP violating terms to be small, the full
list of processes of interest is thus
! W +F f; Zf f; H+F f; Af f; Hf f; hf f :
Here, A denotes the CP odd Higgs, H+ the charged Higgs, and H and h the heavy and
light CP even Higgs bosons, respectively. For charged boson
nal states, f denotes any
fermion doublet component with isospin +1=2, and F the corresponding one with isospin
1=2; for neutral bosons, f can be any SM fermion.
A
Z
H± F¯
W± F¯
f
W±
f
W±
f
W±
F¯
A
Z
and a W boson, mediated by schannel bosons with a mass at the scale of the CP odd Higgs A.
f
F¯
W±
f
F¯
W±
f
F¯
χ˜01
χ˜01
χ˜01
χ˜01
χ˜±n
H±
χ˜±n
W±
χ˜01 f˜¯i
χ˜01
χ˜±n
f
W±
F¯
f
W±
F¯
f
W±
F¯
χ˜01
χ˜01
χ˜01
χ˜01
χ˜01
A
Z
f˜¯
i
In gure 1, we show all contributing Feynman diagrams in a condensed form (note
that some of these diagrams may vanish for speci c combinations of internal and external
particles). For future reference, we follow ref. [18] and refer to the top row of diagrams
as (derived from twobody) schannel processes, and to the bottom row of diagrams as
t=uchannel processes (noting that t and uchannel amplitudes are identical in the v ! 0
limit). Likewise, we denote diagrams of the type that appear in the rst column as virtual
internal bremsstrahlung (VIB), diagrams of the type that appear in the second and third
column as nal state radiation (FSR),4 and diagrams of the type that appear in the last
two columns as initial state radiation (ISR).
We explicitly calculate the full analytical expressions for all these processes in the
limit of vanishing relative velocity of the annihilating neutralino pair, see appendix A.1
for technical details.
We then use the Ward identities in eqs. (2.7) and (2.8) to group
diagrams into gauge invariant sets for the case of vector boson
nal states. In general we
identi ed only two of such invariant sets: those diagrams that are derived from twobody
schannel processes and those that are derived from twobody tchannel processes. In the
limit mA
m
 which is phenomenologically particularly relevant because the observed
4We stress that this distinction between VIB and FSR, while useful for the speci c purpose of our
discussion, is not gauge invariant and exclusively refers to the topology of the involved diagrams. In
particular, it should not be confused with an often used gauge invariant alternative set of de nitions where
FSR refers exclusively to the soft or collinear photons radiated from the nal legs [13, 16, 17], while VIB is
de ned as the di erence between the full amplitude squared and the FSR contribution [13].
f
F¯
W±
f
F¯
W±
f
F¯
W±
Higgs is very SM like  the schannel diagrams however split into two gaugeinvariant
subsets. All diagrams then fall quite neatly into 3 categories: heavy Higgs schannel,
which are the set of diagrams with (at least one) mediator at the mass scale MA (see
gure 2), weakscale schannel, which are the set of diagrams with schannel mediators at
the weak scale (see gure 3), and tchannel, which are the set of diagrams with sfermion
mediators (see
gure 4).5 For Zf f and hf f
nal states the three sets of diagrams can
be obtained analogously: tchannel contributions involve at least one sfermion line, while
the remaining diagrams belong to the schannel category (which can be further split into
subsets involving at least one mediator at scale MA, or none, respectively).
lation rate by a factor of mf2 =m2 is indeed the combination of in principle independent
Yukawa and isospin suppressions. Let us now turn back to this observation and discuss it
in more detail in light of the MSSM, where both mechanisms are still intrinsically linked
because of the connection between gauge symmetry and chiral structure in the MSSM
Lagrangian.
3.2.1
Yukawa suppression
The chiral symmetry of the MSSM Lagrangian is broken by terms proportional to Yukawa
couplings (in order to avoid
avourchanging neutral currents, we assume as usual that
the Aterms are proportional to the Yukawa coupling matrices). Following the general
arguments of section 2.1, any amplitude contributing to
! f f must therefore be
proportional to yf . Within the MSSM the values of yf are functions of tan
but, except for the
top quark, in general so small that this can lead to a suppression of the twobody amplitudes
by many orders of magnitude. From the point of view of the broken theory, this Yukawa
suppression appears to arise from rather di erent types of contributions to the Lagrangian:
i) fermion mass terms
ii) couplings of any of the ve physical Higgs elds to fermions
iii) couplings of fermions to sfermion mass eigenstates (which mix the left and
righthanded elds).
For example, the rst case is relevant for annihilation into fermions via tchannel sfermion
exchange if the sfermion mixing is small (otherwise, the third contribution can dominate
the amplitude), and the second for annihilation via schannel pseudoscalar mediation.
We note that all three interaction types couple left and righthanded states and hence
can ` ip' the helicity of one of the nal state fermions. The helicity combinations that
5We note that for v ! 0 the two schannel ISR diagrams are actually identical, but for clarity we still
include them separately in gures 2 and 3. For v ! 0 and mF ! 0, also the two tchannel ISR diagrams are
important crosscheck of our nal amplitudes, we con rmed analytically that these identities indeed hold.
would result in a chirally symmetric theory, fR;LfR;L, can thus be transformed into those
compatible with the global symmetry requirements outlined in section 2.1, fR;LfL;R.
Traditionally, the notion of this helicity ip is sometimes taken to refer speci cally to the case
(i), in which it is the (kinematic) fermion mass that breaks chiral symmetry in the
Lagrangian. Instead, we associate the e ect directly with the Yukawa couplings in the MSSM
Lagrangian (which of course give rise to the SM fermion masses).
As also discussed in section 2.1, the annihilation process
! f f furthermore violates
I 6= 0, and therefore its amplitude has to vanish in the gauge restoration
! 0. The resulting isospin suppression by a factor v
vEW =m , for heavy
neutralinos, can arise from di erent terms in the Lagrangian of the broken theory:
a) fermion mass terms
b) mixing of di erent gauge multiplets (Bino, Higgsino, Wino) that contribute to the
lightest neutralino mass eigenstate given by eq. (3.1)
c) mixing of left and righthanded sfermion eigenstates.
The structure of the neutralino mass matrix (3.2) indeed con rms that neutralino mixings
vanish for vEW
! 0, as required by SU(2)L invariance. Note that case (a) and (c) are
intrinsically linked to an accompanying chirality violation, since mf / yf vEW and the o
diagonal terms in the sfermion mass matrix are also proportional to yf within the MSSM.
Let us consider as an illustration the t and schannel contributions to
! f f . The
kinematical helicity suppression due to the fermion mass mf is relevant for the tchannel
(sfermion exchange). In this case Yukawa and isospin suppression simply arise from the
two factors in mf / yf vEW (case (a) and (i), respectively). In addition, the Yukawa and
isospin violation can be due to the sfermion mixing (case (c) and (iii)). Indeed, due to
the mixing, a given sfermion mass eigenstate can couple to both left and righthanded
fermions, which then gives rise to the required chirality ip.
For schannel annihilation, on the other hand, the situation is more interesting in
the sense that Yukawa and isospin suppression cannot simply be traced back to the same
origin. For a pseudoscalar Higgs boson A as mediator, e.g., the Yukawa suppression stems
directly from the Yukawa coupling / yf Af f (case (ii)), while the isospin suppression arises
from the neutralino mixing (case b): for pure gauge multiplets the coupling A
would be
forbidden by SU(2)L invariance, and therefore vanishes for vEW ! 0. For a Zboson in the
schannel, the discussion of the limit vEW ! 0 is a bit more involved (see appendix A.3),
but is essentially analogous to the case of an A mediator.
3.3
Yukawa and isospin suppression lifting
In section 2.2, we discussed which threebody nal states
the Yukawa and/or isospin suppression of the process
B is a SM gauge boson or a Higgs boson. This general discussion based on isospin and
! BF f can potentially lift
! f f , for the case in which
FRfL or FLfR
FLfL
FRfR
t=u; s
vEW
vEW
yf
t=u; s
vEW
t=u
vEW
h=H
yf
t=u; s
vEW
t=u; s
vEW
t=u; s
We also indicated whether the process can be realized with the maximal enhancement allowed
by chiral and isospin symmetry in t + u and schannel annihilation processes, respectively. For
the rst two columns we also specify for which neutralino composition (B~ = binolike, W~
=winolike, H~ =Higgsinolike) the maximal enhancement occurs. For the last three columns t + uchannel
processes are possible for B~ or W~ like neutralino as well as mixed H~ =B~ or H~ =W~ , and schannel
processes are possible for mixed H~ =B~ or H~ =W~ . Entries with a dash do not contribute to the order
we are working in (see appendices A.2 and A.3 for details).
chiral symmetry in the limit vEW ! 0 can be extended to the MSSM, as shown in table 2,
by noting that all physical Higgs bosons h; H; A; H
have isospin I = 1=2. Compared to
the gauge restoration limit ZL is given by the Goldstone boson G0 (and hence transforms in
a similar way as the pseudoscalar A). Similar arguments apply to the other Higgs bosons.
In appendix A.2, we consider the full analytic expressions for six di erent mass
hierarchies of particular phenomenological interest and determine for each of the previously
discussed gaugeinvariant subsets of diagrams the leading order in vEW and yf . The result
of this exercise is collected in tables 9{11, where we present the ratio of the leading term
for the threebody amplitude and the corresponding twobody amplitude. This allows us,
as also indicated in table 2, to identify which contributions to the threebody amplitudes
actually realize the suppression lifting that we can maximally expect on the basis of our
general symmetry arguments; the `missing' cases, for which we did not nd a contribution
within the MSSM, are marked by a `'. For a detailed technical discussion of the various
lifting mechanisms, and how they are realized at the level of individual diagrams, we refer
to appendix A.3. We provide a graphical summary in table 3, where we show
representative diagrams that realize the lifting of isospin and/or Yukawa suppression, for the sets
of gauge invariant classes of diagrams that can be discriminated in the gauge restoration
limit (in addition to the three sets discussed before, the tchannel can be split into
contributions that remain nonzero in the limit of pure neutralino states (I), and those that
require neutralino mixing (II)). Isospin suppression can be lifted in all cases by the
emission of longitudinal gauge bosons (here represented by the Goldstone bosons) or a Higgs
boson. Lifting of Yukawa suppression, as well as lifting of both suppression factors, is more
restricted. This can be traced back to basic properties of the unbroken MSSM Lagrangian
and the conservation of JCP = 0 (see appendix A.3 for details), explaining the `missing'
ZT , WT
FLfL, FRfR
χ˜01
χ˜01
χ˜01
χ˜01
χ˜01
χ˜01 gv
χ˜01
χ˜01
χ˜01
χ˜01
g
gv
g
gv
f˜
f˜
f˜
g
g
g
g
g
Yf
G0
A0
Yfv
Yfv
Yf
Yf
f
f¯
f
f¯
f
f¯
f
f¯
f
f¯
χ˜01
χ˜01
χ˜01
χ˜01
f˜
g
g
χ˜01 f˜
Yukawa + Isospin
Isospin
Yukawa
! 0. The rows correspond to the four gaugeinvariant subsets of diagrams
that can be discriminated in this limit (see appendix A.3 for details). The rst column corresponds
to the twobody process, and the other columns show various threebody processes. The diagrams
shown in the second column lift both Yukawa and isospin suppression. The diagrams in the third
column lift only isospin suppression, and in the fourth column only Yukawa suppression. We show
only one representative diagram for each topology (ISR/FSR/VIB) and suppression mechanism.
The coupling factors attached to vertices and mass/mixing insertions give the scaling with yf ,
vEW and g of each diagram (for Bino or Winolike neutralinos; modi cations for Higgsinolike
neutralinos are described in appendix A.3). Note that contributions with WT emitted via ISR
(second column, rst and third row) exist for Wino or Higgsinolike neutralinos; those with ZT
emitted via ISR occur only for a Higgsinolike neutralino.
entries in table 2. Let us also highlight that the classi cation procedure revealed ways
to lift the twobody suppression that have not been pointed out for the MSSM before (in
particular Higgsstrahlung via tchannel ISR and a speci c schannel VIB process, shown
in the last column and second/third row in table 3, respectively).
An additional form of suppression, unrelated to the discussion so far, arises in diagrams that
rely on mixing between neutralinos or contain heavy propagators. This mass suppression
takes the form
X
m =MX, where X is the heavy state in question. In particular, both
HJEP09(217)4
schannel contributions to
! f f and a subset of tchannel contributions  those of
type (II), see appendix A.2  rely on mixing the Bino/Wino with the Higgsino. For
example, for a Bino or Winolike neutralino, the twobody amplitude in the schannel is
suppressed by a factor
2 = m2 = 2 if j j
m . For a Higgsinolike neutralino, on the
other hand, it is suppressed by Mi = m =Mi for Mi
m , where Mi = min(M1; M2) (see
table 12).
These suppression factors of the schannel annihilation can be lifted for the case of a
Wino or Higgsinolike neutralino by the emission of a (transverse) W or Z from one of the
initial neutralino lines (ISR). (The corresponding diagram is illustrated in the third row,
second column of table 3.) Additionally, this threebody process simultaneously lifts both
isospin and Yukawa suppression. It is particularly relevant if the twobody
nal states
W W and ZZ are kinematically forbidden, such that the internal gauge boson is o shell.
This is a special case of the threshold e ects that we turn to next.
3.5
Threshold e ects
A given twobody channel
! AB is strongly phasespace suppressed if the CMS energy
is close to the mass of the nalstate particles, and for 2m
mA + mB the corresponding
partial cross section vanishes completely in the v ! 0 limit. If either A or B are o shell and
decay into much lighter states, however, the phasespace opens up again and thereby
potentially increases even the total twobody annihilation rate signi cantly. For the MSSM, this
is particularly relevant for the W +W
and tt channels, which has previously been studied
for speci c neutralino compositions [46, 47] (for an approximate numerical implementation
in the context of relic density calculations, see [48]). For the processes we are interested in
here, threshold e ects can in general appear for any twoboson
nal states (or tt).
For a more detailed discussion of this e ect, it is useful to rewrite the threebody cross
section as (see e.g. [49, 50])
v2!3 =
=
S
S
4E 1
E 2
4E 1
E 2
Z
Z
jM2!3j2 d 3(P ; p1; p2; p3)
jM2!3j2 d 2(P ; p1; q)
dq2
2
d 2(q; p2; p3) ;
(3.4)
where d n(P ; p1; : : : ; pn) = (2 )4 (4)(P
P pi) Q
d3pi
i (2 )32Ei is the nbody phase space
element, P = p 1 + p 2 the sum of the 4momenta of the annihilating neutralinos and E i
their energy; the pi denote the nalstate momenta. Since q2 = (p2 + p3)2 is timelike, we
will in the following often use the notation q2
m223 instead. For the processes considered
here, cf. eq. (3.3), the symmetry factor S is always 1.6 Furthermore, jMj2 denotes the usual
squared matrix element, averaged over initial spins and summed over nal spins/helicities.
We now assume that the amplitude is dominated by a resonant, almost onshell internal
propagator that decays into particles 2 and 3, and hence carries momentum q. For a
resonance R with mass M , width , and spin 1, 1=2, or 0, respectively, we then have
M2!3 =
1
m223
M 2 + iM
<
:
>
>
8
>> M(2q!) 2( g
M(2q!) 2(=q + M )M1!2
(q)
M2!2M1!2
(q)
+ q q =M 2)M1!2 vector
fermion
scalar
(3.5)
where M2!2 (M1!2) is the matrix element for
! p1q (R
! p2p3), up to polarization
vectors or spinors for the `external' particle R (as indicated by the superscript q).
The decisive observation is now that R
2
jM1!2j d 2(q; p2; p3) must be independent of
the polarization state of R once all the nal state polarizations are summed over. This is
familiar from onshell momenta q  the total (but not di erential) decay rate of a particle
is independent of its polarization state  but holds more generally for timelike initial
momenta q [50]. As long as the full phasespace integral is performed (see section 4.2 for
how to treat di erential cross sections), one may thus conveniently replace the correlated
polarization or spin structure of eq. (3.5) with an unpolarized sum:
M(2q!) 2( g
+ q q =M 2)M1!2
(q) 2
X
(q)
M2!2
M(2q!) 2(=q + M )M1!2
(q) 2
M(2q!) 2ususM1!2
(q) 2
=
1
=
In this way, we can independently of the spin of R replace
jM2!3j2 ! (m223
2
in eq. (3.4) which, for v ! 0, leads to
v2re!s 3 = S
Z (2m
m1)2 dm223
(m2+m3)2
(m223
m23
M 2)2 + M 2 2 R!23 fv
~
!1R :
6In general, if some of the nal state particles are of identical type, con gurations that di er only by
exchanging these particles should be counted only once in the phase space integration. Since this will be
convenient later on, we thus use a convention where one integrates over all of the phase space as if all
particles were distinct, and then correct for the corresponding overcounting by a symmetry factor S. It is
S = 1 if all nalstate particles are distinct, and S = 1=2 (S = 1=6) if two (all three) of them are identical.
(3.6)
(3.7)
(3.8)
(3.9)
Here, the decay rate of the o shell resonance in the frame where q = (m23; 0) is given by
and the cross section for the annihilation into an o shell resonance is given by
~
R!23
v
f
!1R
S23 Z
2m23
S1R Z
P 2
2
2
16
S1R
128
jM1!2j d 2(q; p2; p3) =
In the last step we performed the phasespace integral explicitly by using the fact that for
v ! 0 the annihilation process is kinematically the same as a pseudoscalar decay, implying
that jMj cannot have any angular dependence.7 Eq. (3.9) will thus continue to hold for
general swave annihilation, provided one replaces 4m
! s in eq. (3.11). The squared
matrix elements are here again summed (averaged) over nal (initial) spins/helicities,
leading to an overall symmetry factor of S = S=(S1RS23) (with S1R; S23 de ned in accordance
with footnote 6).
We note that eq. (3.9) can be signi cantly simpli ed by a few wellmotivated
assumptions. Concretely, let us assume the o shell particle to decay to massless nal states, m2 =
m3 = 0, and jM1!2j2 / M 2 close to the threshold; this implies ~R!23 = (m23=M ) R!23.
We also introduce a reduced cross section
( v)red
( v)
!1R= n+1=2(1; 1; R) ;
(3.13)
with R
m223=s and 1
m21=s, allowing for the twobody cross section close to threshold
to be suppressed not only by a phasespace factor (n = 0), but by an additional such factor
from the matrix element itself (as e.g. in the example of Higgsino annihilation below, for
which we have n = 1). By de nition, ( v)red thus remains
nite both above and below
the threshold. Assuming ( v)red to be independent of m23 close to threshold, eq. (3.9)
simpli es to
v
!1R ' S( v)red
Z max d
0
(
1)2 + 2
n+1=2(1; 1;
R) ;
(3.14)
where
max = ( s
p
m1)2=m2R and
the sense that the threshold correction can be directly estimated for any given twobody
cross section (i.e. without rst having to compute ~v or ~).
R=M . This expression is modelindependent in
As an illustrative and concrete example, let us consider the process the limit of pure Higgsino DM. For simplicity, we assume that sleptons are much heavier
7For this reason, the result takes the same form as for o shell decays [51, 52], suggesting a
straightforward generalization to 4body nal states dominated by the annihilation into two o shell particles:
v2re!s 4 = S
Z dm122 dm324
m12
m23
(m212 MR21 )2 +MR21 2R1 (m223 MR22 )2 +MR22 2R2 ~R1!12 ~R2!34 fv !R1R2 :
! W e+ e in
(3.12)
mχ±=200 GeV
W
Γ
±
W
m
χ
[
! W e+ . For the latter process, we show the cross section divided by the branching fraction
W !e = W ' 1=9 (solid lines). For comparison, we also include the modelindependent estimate of
(W +) (dotted lines). For m
. mW , the threebody cross section is clearly
larger than the lowestorder result; above the threshold, on the other hand, the two agree exactly.
! W +W , compared to
than the neutralino, such that the only contributing diagrams are of the V = W
ISR type,
with a virtual Higgsinolike chargino and a resonance R = W + . In this limit, we nd
v
f
!W W
16m4
8m2 (m2W + m223) + (m2W
2m2 + 2m2~+
m2W
m223
We calculate the full threebody cross section as derived in appendix A.1, in the pure
Higgsino limit, and then compare it to the result given in eq. (3.9). As shown in
gure 5, we obtain excellent agreement even though both the directly involved amplitudes
and the numerical phasespace integrations are very di erent in nature (the two results for
the threebody cross section, shown as solid lines, lie exactly on top of each other). This
should of course be expected for a process which by construction only receives
contributions from an o shell nalstate particle, but we stress that eq. (3.9) is in general much
simpler to calculate in praxis for such cases. For comparison, we also indicate (with dotted
lines) the modelindependent result given in eq. (3.14); as one can see, even this simpli ed
expression provides an excellent approximation to the full result.
Most importantly, our example illustrates the much more general point that a
threebody process around or below the kinematic threshold of a large twobody process can be
signi cantly enhanced over the total annihilation rate at lowest order. Above the threshold
and rescaled to the relevant branching ratio for the decay of the resonance R, on the other
(3.15)
(3.16)
hand, the threebody cross section for a process
! 1R; R ! 23 equals almost exactly
the twobody result  an e ect which we will discuss in detail in the next section.
4
Double counting issues
We now turn to double counting issues related to unstable nalstate particles. If the
nal state of a twobody annihilation process undergoes a subsequent 1 ! 2 decay, in
particular, this can also be viewed as a threebody process with the unstable particle (the
resonance, in our wording) as an intermediate state. While we discussed the situation below
the kinematic threshold for the production of the unstable particle in section 3.5 as a way
of enhancing the total cross section, we are here interested in the kinematic region above
the threshold. As before, this is relevant for all massive diboson as well as tt nal states
considered here.
One possibility to avoid overcounting identical kinematic con gurations when adding
twobody and threebody processes would be to altogether disregard the former for massive
diboson or tt nal states. Interferences between (nearly) on and o shell contributions to
the amplitude would then be correctly accounted for, as well as the impact of the spin of
the resonance. However, this procedure has several drawbacks on a practical level, and
furthermore turns out to be incorrect for twobody processes with identical particles in the
nal state (such as e.g.
! ZZ), as will be discussed in more detail below. We therefore
prefer to explicitly subtract onshell contributions to the threebody processes, which allows
us to keep most of the advantages of the full threebody computation while correctly taking
into account all symmetry factors. In the following we describe this procedure in more detail
for both the total cross section and the di erential yield of e.g. gamma rays.
4.1
Narrow width approximation and total cross section
For threebody processes dominated by an on or o shell resonance, the total cross section
can be written as in eq. (3.9). If the intermediate particle corresponds to a nearly onshell
resonance with
M , furthermore, the BreitWigner propagator can be approximated as 1
(m223
M 2)2 + M 2 2 ! M
(m223
M 2) :
This narrowwidth approximation (NWA) yields the onshell contribution of the resonance
R, and we denote the corresponding, approximated cross section by vNW A. Strictly
speaking, for the approximation to work well, the kinematic boundaries have to be su ciently far
away from the pole, jm223
M 2j
phasespace factors apart from the BreitWigner propagator should be smooth functions of
m223 in the vicinity of the pole, which we assume in the following. With this replacement
in eq. (3.9), we immediately recover the wellknown result
M , and all contributions from the matrix element and
v2N!W3A = S
v
!1;R
BRR!23 ;
S2!2=P 2 R d 2jMpj2 is the twobody cross section.
cles 2 and 3, R!23 = S1!2=(2M ) R d 2jMdj2 is the partial decay width, and v
where BRR!23 =
R!23= is the branching ratio for the resonance R to decay into
parti!1R =
(4.1)
(4.2)
For Higgs nal states, the summation over polarizations is absent, and we de ne the
corresponding ratios RLR=RL and RLL=RR analogously, corresponding to hfRfL + hfLfR and
hfLfL + hfRfR
nal states, respectively.
We then start from our full result for the helicity amplitudes, using the explicit
representations of the generic couplings and mass matrices that appear there, and expand them
up to O( v2). Note that the limit v ! 0 implies in particular that we expand in the fermion
mass mf / yf vEW and in gauge boson masses mW=Z / gvEW . In order to simplify the
resulting analytic expressions, we set all sfermion masses equal to the neutralino mass, noting
that larger sfermion masses would suppress t=uchannel rates relatively strongly because
v2t!c3hannel
/ mf~8(mf~4) for Bino (Higgsino/Wino)like neutralinos, as discussed
previously [
31, 33, 36, 37, 92
]. Furthermore, we use the notation BF f where for neutral bosons
(B = Z; h) the
nal state fermion types are identical, F = f , while for charged bosons
we adopt in the following the convention that f denotes the uptype fermion (e.g. the top
quark in
! W bt). In these cases, we keep for simplicity only the dependence on the
Yukawa coupling of the uptype fermion, and set the other one to zero. We furthermore consider six distinct scenarios describing the dominant neutralino composition, which result from di erent assumptions about the involved mass hierarchies and which are of particular phenomenological interest:
Higgsino DM, with small Bino admixture (
Higgsino DM, with small Wino admixture (
Bino DM, with small Wino admixture (M1
M2;
Bino DM, with small Higgsino admixture (M1
Wino DM, with small Bino admixture (M2
M1;
Wino DM, with small Higgsino admixture (M2
M2; M1 ! 1)
M1; M2 ! 1)
! 1)
! 1)
; M2 ! 1)
; M1 ! 1)
From the Bino, Wino and Higgsino mass parameters M1, M2 and , we de ne the
dimensionless mass suppression factors M1
m =M1, M2
m =M2 and
m = . For all
six scenarios listed above, we expand the amplitude ratios to leading order in these mass
suppression factors. E ectively, the neutralino mixing between either Bino or Wino and
Higgsino then becomes a perturbative `mass insertion' / gvEW represented by the
respective o diagonal entries in the mass matrix of eq. (3.2). Furthermore, for de niteness, we
also expand to linear order in A
m =MA, i.e. we work in the decoupling limit where
the heavy Higgs states are much heavier than the neutralino or SMlike Higgs boson. We
note that it is straightforward to generalize these results, and our numerical results anyway
include all MSSM Higgs bosons and are valid for arbitrary mass hierarchies.
To lowest order in the expansion parameters de ned above, isospin and fermion
chirality have to be conserved in all interaction vertices (assuming that the mass splitting
between M1, M2 and j j is large compared to gvEW ). One of the implications, as it turns
out, is that the gaugeinvariant subset of tchannel diagrams discussed in section 3.1 can
be further split into two separate gaugeinvariant sets. The rst, which we will denote by
(I), does not contain any neutralino mixing insertion / gvEW , and would hence contribute
even in the limit of a pure neutralino state. The set of diagrams that contain at least one
such insertion (denoted by (II)), on the other hand, require a mixing in the neutralino
sector (just like is the case for all schannel diagrams).
In tables 9{11, we show the results of this expansion for the helicitysummed ratios R
that we have introduced in eq. (A.16), where the di erent tables correspond to the three
types of nal states (W F f , Zf f , and hf f , respectively). Each table contains the results
for all six mass hierarchy scenarios speci ed above, broken down to contributions from
each set of gaugeinvariant diagrams.19 For the sake of the presentation, we keep only
contributions that lift the isospin or Yukawa suppression of the corresponding twobody
process (or both). In particular, as apparent from table 2, the ratio RLTR=RL cannot lift any
of these suppressions, and is therefore not included in tables 9{11. Furthermore, for each of
the gaugeinvariant sets of diagrams, we include only those amplitude ratios that actually
do lift at least one of the suppression factors. For the remaining entries, a `0' indicates
threebody amplitudes that vanish to the order we consider, while for entries containing a
` ' both 2 and threebody amplitudes vanish.20 The twobody amplitudes, nally, are for
convenience summarized in table 12.
19We checked explicitly (up to O( v2)) that the Ward identities, eqs. (2.7), (2.8) are satis ed for each set
separately. Note that we use BreitWigner widths in the amplitudes, and while they break gauge invariance
at O( v), they do not contribute to the amplitudes as v ! 0.
20While in this case the ratio would be formally illde ned, we only identi ed one example where the
twobody amplitude vanishes while the threebody amplitude does not, marked with a ( ). We note that
the relevant process,
! W W
! W F f for a Winolike neutralino, is phenomenologically not important
because for m
> MW it is largely captured by annihilation into W W , while Winolike neutralinos with
m < MW are practically excluded.
)
I
I
r
o
t
a
i
d
e
l
e
n Z
n
m
a
h
c
s
f
f
h
!
+ 2 0
0
L R
f f
L
)
R L 2
f f ;x v
L R
1
x 2
f f (
h h E
)
2
;x v
1
x 2
(
~
L
2
R J x
2
x
)
1
2
x
+
1
x
+
1
x
(
f
y
2
)
1
2
x
v
2 1
x x
+ v
x1 2
p
2
)J +1
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HJEP09(217)4
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In order to assess the parametric enhancement of threebody over twobody processes,
it is su cient to consider the amplitude ratios just presented, and we will continue with
a more detailed discussion of the various lifting mechanisms at the level of individual
diagrams in the following subsection A.3. Before doing so, let us brie y remark that the
corresponding cross section ratio for
obtained by
! BF f , normalized to the one for
! f f , is
1
v2!2
d( v)2!3 =
1
are the dimensionless fermion energies of the threebody
nal state.
Using the results from tables 9{11, one can thus obtain the contribution to this ratio from
each of the gaugeinvariant subsets of diagrams separately. In the limit of massless nal
state particles, the integration ranges are 0 < x1 < 1 and 1
x1 < x2 < 1, implying that
some of these integrations become logarithmically divergent. This is an expected artefact of
the expansion in v and, in practice, the corresponding infrared divergent contributions are
cut o by the nonzero mass of the vector boson. Throughout this work, we assume that the
resulting logarithmic enhancement O( ln2(EB=ps)) can be treated perturbatively down
to the infrared cuto
mass of roughly m
EB
mB
gvEW . This imposes an upper limit on the neutralino
O(gvEW e =g)
O(10) TeV. If one is interested in higher masses,
it would be interesting to apply the resummation methods discussed e.g. in refs. [55{57].
On the other hand, we stress that the logarithmic sensitivity to ln2(g v) does not spoil the
power counting arguments related to lifting of isospin suppression factors, since the latter
is described by powers vn of v. In our numerical results, we fully take into account the
masses of all annihilation products.
A.3
Suppression lifting from individual diagrams
It is rather illustrative to re ect the results of the previous subsection at the level of
individual diagrams. In table 3, displayed for clarity already in the main text (see section 3.3),
we therefore organize all relevant amplitudes in a large table, with the four rows
corresponding to the four gaugeinvariant subsets. For each type of diagram, and assuming a
Bino or Winolike neutralino, we furthermore explicitly indicate the scaling with the gauge
coupling g, the Yukawa coupling yf , and the vev vEW (we comment on the Higgsinolike
case below). Let us start our discussion with the rst column, which contains the diagrams
contributing to the twobody process
/ g2yf vEW , but the origin di ers:
! f f . As expected, all these amplitudes scale as
tchannel I. The factor yf vEW enters either via the chirality ip of one of the nalstate
fermions, or via a L=R mixing insertion of the sfermion (for brevity, we show only
one representative diagram in table 3 for each of these cases).
tchannel II. The factor vEW enters via the gaugino/Higgsino mixing insertion on one of
the initial lines, and the Yukawa suppression enters via the Higgsinosfermionfermion
coupling.
schannel EW. The schannel with electroweakscale mediator corresponds to the
Zexchange diagram mentioned earlier. In the swave limit, and from the perspective of
the unbroken theory, this diagram is represented by the exchange of the pseudoscalar
Goldstone boson G0. The factor vEW arises from the gaugino/Higgsino mixing, and
the Yukawa coupling from the Yukawa interaction G0f f .
schannel MA. This case is similar to the previous one, except that the mediator is
replaced by the (physical) heavy pseudoscalar Higgs A.
Let us now turn our discussion to the remaining columns of table 3, which contain
all relevant threebody processes. Here, the second column shows representative Feynman
diagrams that lead to a lifting of both isospin and Yukawa suppression, while the third and
fourth column show diagrams that lift only one of them, respectively:
HJEP09(217)4
Lifting of Yukawa and isospin suppression. Both suppression factors can be lifted
only for two of the gaugeinvariant sets of diagrams (tI and sEW). In the former
case, a transverse ZT or WT is emitted from either fermion line in the nal state, from
the sfermion line, or from the initial lines (this last case cannot occur in the Binolike
case). We remark that FSR can only lift the helicity suppression if the virtual fermion
is strongly o shell, i.e. not for soft and collinear photons (which are sometimes
dened as FSR, see footnote 4). In the schannel case, the diagrams can be thought of
as an annihilation
! W W , with subsequent decay of W
(see section 3.5 for a
discussion of such o shell internal states). It is impossible to lift both suppression
factors for the other two classes: for tII, this would require a gauginoHiggsinoW=Z
vertex, which is absent for vEW ! 0. The same applies for sMA, noting in addition
that the Af f coupling requires also the presence of a Yukawa coupling.
Lifting of only isospin suppression. The isospin suppression can be lifted for all
four subsets, by replacing the insertion of vEW
within the twobody amplitude
by the emission of a Higgs boson or a Goldstone boson, respectively.
Note
that for the set tI this amounts to replacing the fermion mass insertion by a
fermionfermionHiggs/Goldstone coupling (or replacing the sfermion L=R mixing
insertion by a sfermionsfermionHiggs/Goldstone coupling, respectively). For all
other sets one replaces the gauginoHiggsino mixing insertion in the initial line by a
gauginoHiggsinoHiggs/Goldstone vertex. For the schannel, the diagrams can also
be thought of as an annihilation into a pair of scalars, with subsequent decay of one
of them. This mechanism of suppression lifting is very general, and appears for all
gauge invariant subsets of diagrams as well as for all nal states (involving W=Z or
a Higgs boson). We expect it to be relevant especially for heavy neutralino masses.
Lifting of only Yukawa suppression. This case is in some sense the most di cult to
realize. The reason is that it requires a Higgs (or Goldstone) boson in the nal state,
and therefore only diagrams where the nalstate boson does not couple directly to
the nalstate fermions can potentially contribute in the limit yf ! 0. We identi ed
three such processes, shown in the last column in table 3: for tI, the Higgs (or charged
Goldstone boson; note that there is no sfermionsfermionG0 vertex for yf ! 0) can
be emitted from the sfermion line in the tchannel, i.e. via VIB. The corresponding
vertex is derived from a fourscalar sfermionsfermionHiggsHiggs interaction,
involving the full Higgs doublets. This coupling leads to the required vertices at O(vEW ),
and scales with g2 for yf ! 0 within the MSSM (see refs. [33] and [20] for a discussion
within a toy model for the Goldstone and Higgsemission, respectively). In addition,
for tII, the Higgs can be emitted via ISR (second row, last column of table 3).
While this contribution lifts Yukawa suppression, it is suppressed compared to the
twobody process for a large mass hierarchy between gaugino and Higgsino mass
parameters; we nevertheless kept this contribution, because the former e ect can easily
compensate for the latter. Finally, for the sEW case, the Higgs can be emitted from
the schannel mediator via a GoldstoneHiggsZ coupling (third row, last column in
table 3). Note that this mechanism is distinct from the one discussed in [38], and
that the toymodel discussed there cannot be realized within the MSSM. To the best
of our knowledge, both the tchannel ISR and the schannel Higgstrahlung processes
that we identi ed within the MSSM have not been discussed before.
One can understand the diagrams that lift Yukawa or isospin suppression as shown
in table 3 based on basic properties of the unbroken MSSM Lagrangian, as well as the
symmetry requirement JCP = 0 of the swave initial state. For example, mixing insertions
/ gvEW of the neutralino line can turn a Bino into a Higgsino, but not into a Wino. In
addition, the Higgsino coupling to fermion/sfermion pairs is proportional to the Yukawa
coupling, while the corresponding coupling for Bino and Winolike neutralinos involves a
gauge coupling and is therefore generally much less suppressed (except for the top quark).
One slightly more involved example is the diagram in the last column of the rst row.
For
nal states involving a longitudinal WL, the corresponding sfermion vertex derives
from the interaction term / g2(f~Ly H)(Hyf~L) present for sfermion
doublet under SU(2)L. After inserting the decomposition H = (G+; (vEW + h + iG0=p2))
elds that transform as
of the SMlike Higgs doublet one easily veri es that at linear order in vEW one obtains
a sfermion coupling to G
and h, but not to G0, which explains why no longitudinal ZL
boson can be produced in this case. The Higgs nal state also receives a further contribution
from the interaction term / HyHf~yf~, which exists for all (left and right) sfermion
elds.
Furthermore, for the schannel processes of the type
! hB
! hf f that give a nonzero
contribution in the swave limit, the mediator B is a pseudoscalar or transverse vector (i.e.
G0; A0, ZT ), while for
! G0B
! G0f f , B is a scalar (i.e. h; H0). This is consistent
with the odd CP parity of the initial state.
Note that the above arguments are only valid when expanding around the unbroken
theory, and representing longitudinal degrees of freedom by Goldstone bosons. In fact,
within the broken theory, analogous arguments would be hampered by large cancellations
that occur among individual diagrams, and that make the power counting less transparent.
Nevertheless, we carefully cross checked that all these arguments can indeed be reproduced
when using the full matrix elements within the broken theory, and expanding the sum of
all diagrams within a gauge invariant subset for heavy neutralino mass.
While the discussion above assumed a gauginolike neutralino, the case of a
Higgsinolike neutralino is very similar. For the third and fourth row in table 3, in particular, nothing
changes except that the incoming neutralino is now a Higgsino in the limit vEW ! 0, and
the insertion / gvEW denotes mixing with either a Bino or Wino (in addition, both ZT
and WT ISR is possible, while only WT ISR is possible for Winolike neutralinos). The
same applies to the second line, after interchanging the label of g and yf on the vertices
involving a sfermion in all diagrams in the rst and second column (this does not a ect the
overall scaling of the amplitude), while the diagram in the last column would receive an
additional yf2 suppression. For the rst row, the two neutralinosfermionfermion vertices
scale with yf instead of g in all diagrams. Thus, this class is additionally suppressed by a
factor yf2 compared to the other subsets. Nevertheless, for completeness, we kept this case
because the threebody processes can lift the additional suppression factors yf vEW of the
twobody amplitude in the same way as for a gauginolike neutralino.
In summary, we con rmed the general symmetry arguments outlined in section 2.2 for
the MSSM and explicitly identi ed the contributions to the threebody amplitudes that
realize the suppression lifting, focussing on
nal states containing (tranverse or
longitudinal) gauge bosons as well as the SMlike Higgs boson. By expanding the full amplitudes in
various limits that correspond to Bino, Wino or Higgsinolike neutralino, respectively, we
nd that (almost) all of the possibilities allowed by symmetries are realized. The cases for
which we did not nd a contribution within the MSSM are marked by a `' in table 2. For
processes involving W bosons and purely righthanded fermions an additional suppression
arises that can be traced back to the chiral structure of the SU(2)L interaction. For
processes involving ZL (represented by G0) or A, and fermions of equal chirality, on the
other hand, lifting of Yukawa suppression would require that the amplitude does not
contain Yukawa interaction vertices. In addition, vertices such as sfermionsfermionG0=A
are absent for yf ! 0 (as required by CP invariance), such that a tchannel process
analogous to the one in the rst row, last column of gure 3 does not exist. For the
schannel, the symmetries of the initial state would require a CPeven mediator if the G0 or
A was emitted via ISR. Within the MSSM, only the Higgs bosons are available. However,
their coupling to fermions necessarily involve a Yukawa coupling, such that Yukawa
suppression cannot be lifted in this speci c process. Similarly, one can convince oneself
that the schannel VIB process (3rd row, 4th column of gure 3) as well as the remaining
tchannel process (2nd row, 4th column) cannot occur when replacing h ! G0; A.
B
Numerical implementation
For each Feynman diagram, we have implemented the full analytical expressions for the
helicity amplitudes in DarkSUSY [40].
We numerically sum over these contributions to
obtain the total amplitude for a given helicity con guration, M
appendix A.1. Di erential and partial cross sections are computed according to eq. (A.13),
by numerically integrating over the energies of the
nal state particles; for consistency
checks, this can be done for any pair of energies and in any speci ed order. In order to
(h; )
!F fX , as introduced in
HJEP09(217)4
improve convergence and accuracy of the numerical integrations, we use taylored integration
routines that make use of the known locations of kinematic resonances [87].
For the total cross sections, we have explicitly implemented the NWA approximations
contained in eqs. (4.3){(4.9). We have extensively checked our code, and hence also the
prescription of subtracting the NWA contribution detailed above, by comparing the total cross
section de ned in eq. (4.10), on a channelbychannel basis and for various SUSY models,
with numerical results obtained with CalcHEP [93].21 For all models, and all annihilation
channels, we nd remarkable agreement. We also checked agreement for individual classes
of diagrams (s=tuchannel, ISR/FSR/VIB) as classi ed in section 3.1. Let us stress that
in terms of computation time the implementation via helicity amplitudes, together with
HJEP09(217)4
the taylored integration routines, is less expensive compared to the evaluation of squared
matrix elements via Monte Carlo integration as implemented in CalcHEP. This is especially
signi cant for the threebody processes to which a large number of diagrams contribute,
and for which the di erence in computation times amounts to several orders of magnitude
in the speci c kinematic limit we are interested in here.
For the yields of stable particles, we have implemented the procedure described in
section 4.2, using unpolarized yields for decaying particles given that these are the only
ones that are currently available in DarkSUSY [95]. As discussed, as long as the total yields
(i.e. summed over all channels) are concerned, our prescription still captures any double
counting. We note that extending our implementation to fully polarized yields will be
straightforward for future work, given the results provided in section 4.2 and the helicity
amplitudes reported in appendix A.
Let us mention a few of the extensive numerical checks that we performed to test
the yield implementation. We considered, in particular, models for which the threebody
annihilation is dominated by an almost onshell intermediate resonance. In this case, the
21We compared our implementation of threebody cross sections based on DarkSUSY 5.1 with CalcHEP
3.4 [93]. In particular, we adapted the ewsbMSSM implementation of CalcHEP to compute the spectrum
from a given set of pMSSM input parameters at scale Q = MZ (except for MA which is the pole mass)
using SoftSusy 3.4 [94]. The Susy les Houches output le written by SoftSusy is then used as input for
DarkSUSY via the slha interface. In order to be able to directly compare the output it is necessary to
adapt various routines in order to match the conventions. Apart from making sure that all SM input
parameters agree (we used mb = 4:92 GeV, sin( W ) = 0:47162,
W = 2:07 GeV, t = 2:0 GeV), we made
the following changes for the purpose of cross checking: for CalcHEP, we switched o the running bottom
mass (dMbOn=0) and used unitary gauge (for the comparison on a diagrambydiagram basis; only the sum
is gaugeindependent). For DarkSUSY , the Yukawa couplings are by default read in from the blocks YU,
YE and YD in the slha le. For the purpose of comparison, it is convenient to
x the Yukawa couplings at
yi = mi=v, especially for the top. Therefore, we commented out the corresponding lines in dsfromslha.f.
Additionally, in su/dssuconst yukawa running.f, we commented out the running Yukawas, such that the
default Yukawa couplings, which are simply related to the (onshell) masses, are used. In addition, the
call to dshigwid() was commented out in dsfomslha.f in order to avoid a rescaling of Higgs couplings
that takes certain NLO corrections into account. For the purpose of comparison, it is more convenient to
have treelevel couplings. In addition, we then set the Higgs widths to a common value in both programs.
Finally, we set the rst and second generation quark masses to zero and the CKM mixing matrix to unity
in order to match the conventions of the ewsbMSSM model implemented in CalcHEP. We veri ed that the
conventions agree by comparing also the twobody cross sections for all channels allowed at swave, for
which we nd perfect agreement after the changes described above.
subtraction procedure described in section 4.2 is expected to lead to a large cancellation
between the full threebody contribution and the NWA term. We explicitly veri ed this
cancellation for all yields of stable particles, and over the full energy range. The cancellation
amounts to several orders of magnitude in speci c cases, and therefore provides a robust
check of the implementation. In addition, we also veri ed that the yields obtained from
all of the models contained in our MSSM scan results pass a number of checks (e.g. yields
within an expected range at E > m =2 and E > m =10). Finally, we also considered
threebody
nal states that contain directly one or more stable particles (such as e.g.
! W e ). In this case, we veri ed that the neutrino and positron spectra match the
analytical result discussed in appendix C for speci c models for which this nal state is
dominantly produced by an intermediate W resonance.
C
Spin correlations of decaying resonances
In section 4, we discussed how to subtract double counting due to onshell intermediate
states (`resonances') contributing to threebody annihilation processes. If the resonance
carries a spin, the spectrum of
nal state particles depends on how much the various
helicity states of the resonance contribute. In section 4 we argued that for annihilation of
Majorana fermions in the swave limit, CP and angular momentum conservation uniquely
determine the helicity of all possible intermediate states that can contribute to the
threebody processes considered here. Here we present a formal derivation of this result, based
on a description that would in principle allow us to treat also more general cases.
In full generality, several helicity states of the resonance contribute to the amplitude,
and can also interfere with each other when taking the absolute value squared. As a starting
point we consider the example
! HW ! Hf F . We are interested in the contribution
from the onshell intermediate W boson. The full matrix element squared can then be
written in the form
(Ms1;s2)1!2 = us1 (p1)(gPL
)vs2 (p2)
jMresj
2
X
s1;s2
X
M2!2
(Ms1;s2)1!2
2
where we indicated explicitly the summation over the
and the polarization states of the internal W . To extract the onshell contribution in
the narrowwidth limit we assume that the momentum q of the W is (almost) onshell,
q2 ' M W2 . This implies that the kinematics of the H and W momenta is identical to the
twobody annihilation. The rst term inside the square contains the helicity amplitude for
the twobody part,
M2!2
M2!2
:
q
For concreteness we can take the momentum of the W to be along the zaxis, q
=
(Eq; 0; 0; jqj), where Eq =
jqj2 + M W2 and jqj is determined by the neutralino, W and
Higgs mass via the twobody kinematics (identical to jp3j, see (C.22) below). The decay
W ! f F gives (fermion momenta p1 and p2)
(C.1)
(C.2)
(C.3)
Inserting these into the resonant matrix element, and writing out the square gives after
some renaming of indices
jMresj =
2
X (Ms1s2 )1!2(Ms1s2 )1!2
where
X
; 0
X
; 0
0
0
M2!2M2!2
M2!2M2!2
s1;s2
s1;s2
g2 tr((p=2
2g2 (p2 p1
mF )
PR(p=1 + mf )PL
p2 p1 )
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
(C.9)
(C.11)
(C.12)
(C.13)
The decorrelationapproximation (i.e. the replacement of the matrix element by the product
of two matrix elements with independent summation over spin states, (3.6)) would be
obtained by replacing
. To compute it, one can use the
D 0 ! D 0
decorrelated
1
3
X
= (0; 1; i; 0)=p2;
0 as well as P
0 = (jqj; 0; 0; Eq)=MW ;
g
+ q q =M W2 .
helicity basis
which ful ll
0
axis of the W ),
A convenient frame to evaluate it is the rest frame of the W , obtained by boosting
along the z direction. In this frame 0 = (0; 0; 0; 1). The momenta of the fermions can
be parameterized by the angle w.r.t. to the zaxis (which is singled out as the polarization
p1 = (Ep1 ; 0; jp1j sin ; jp1j cos );
p2 = (MW
Ep1 ; 0; jp1j sin ; jp1j cos )
(C.10)
where jp1j and Ep1 =
qjp1j2 + mf2 are the momentum and energy of f in the W rest
frame, see (C.22). Inserting this and evaluating the trace yields an explicit expression for
D 0 = D 0 ( ) in terms of . Using that the polarization vectors have zero temporal
component in the basis we are working in, and that p~2 =
p~1,
D 0 ( ) = 2g2 ( 2(~ 0 p~1) (~ p~1) +
0 p1 p2 + iMW (~ 0
~ ) p~1)
X (Ms1s2 )1!2(Md;s1s2 )1!2
HJEP09(217)4
Now one can use p1 p2 = (M W2
m2
f
m2F )=2 and
~0p~1 = jp1j cos ;
~+ = +i~0;
~ p~1 =
i jpp1j sin
2
form
where
d( vHfF )NW A =
(2 )3 16(2M )2 jMresj MW W 2MW
1
jp1jjp3jd cos
To obtain the di . cross section, we use the representation of the phase space in the
d( vHfF ) = d j4MMj2 =
2
(2 )5 16(2M )2 jMj jp1jjp3jdm12d 1
d 3
1
where p3 is the Higgs momentum, and m212 = q2 the resonant momentum. Now we can do
an approximation where we replace
2
2
jMj ! jMresj MW W
M W2 )
but keep the fully correlated matrix element jMresj .
By integrating over dm12 =
dm212=(2m12) = dq2=(2MW ), and doing the trivial Higgs angle d 3 and d 1 integrals, one
obtains the di . cross section w.r.t. to the angle
of the fermion f and the polarization
axis of the W boson in the backtoback system,
D
D
D 0 =
D00 = 2g2(p1 p2
= 2g2(p1 p2
= 2g2jp1j2 sin2
2jp1j2 cos2 )
jp1j2 sin2
MW jp1j cos )
ip2g2jp1j sin (MW
2jp1j cos ) = D0 :
One can check that the average over the diagonal contributions corresponds to the usual
unpolarized decay matrix element,
jM1!2j2 =
3
D
1 g2(6p1 p2
4jp1j2) = g2
M W2
mf2
m2F
4
3 jp1j2
jp1j =
jp3j =
[(M W2
(mf + mF )2)(M W2
(mf
mF )2)]1=2
2MW
One can rewrite this expression, using that the twototwo and W decay rate are given by
vHW =
W !fF =
jp3j
X
1
8
8 (2M )2
1 jp1j
M W2 jM1!2j
jM2!2j
2
2
where M2!2 is the helicity amplitude and jM1!2j2 is the usual summed/averaged decay
matrix element. Expressed in terms of the matrix introduced above, jM1!2j = 13 P
D
2
2
2
1
1
2
4M
(C.14)
(C.15)
(C.16)
(C.17)
(C.18)
(C.19)
(C.20)
(C.21)
(C.22)
(C.23)
(C.24)
(C.25)
HJEP09(217)4
The dependence on the angle cancels in this sum. Then,
d( vHfF )NW A
d cos
vHW
W !fF
W
vHW BR(W ! f F )
2(P
2
jM2!2j2)( 13 P
F
!HW !HfF ( )
D
)
(C.26)
(C.27)
(C.28)
(C.29)
where we have de ned the function F which characterizes the angular dependence. Using
also (C.4) for Mres, one can write it as
F
!HW !HfF ( ) =
P ; 0 M2!2M2!2
0
2(P
jM2!2j2)( 13 P
)
If one would replace the matrix D 0 by the decorrelated approximation (C.8), the last
term becomes constant
F
!HW !HfF ( )
D!Ddecorrelated =
1
2
D
:
o diagonal entries.
for D 0 , this imples that
F
!HW !HfF ( ) =
Integrating over the angle d cos (which yields a factor 2), one then recovers the familiar
relation for the NWA of the total cross section. However, in general the matrix D 0 di ers
from the decorrelated approximation, and has a nontrivial angular dependence as well as
For Majorana DM annihilation into a scalar and a vector, only the longitudinal
polar= 1
ization contributes to the swave, i.e. M2!2 ! 0 for v ! 0. Using the explicit expression
D00( )
2 13 P
D
M W2
m2
m2F
2(M W2
m2
m2F
4jp1j2 cos2
3 sin2
4
; (C.30)
where the last expression applies for massless fermions. This corresponds to the decay
spectrum of a longitudinally polarized W boson.
The integral of this expression over
d cos coincides with the decorrelated case. Therefore, the result for the total cross section
in the NWA is nevertheless accurate, with error governed by
W =MW , as expected.
Instead of the angle
one can use the energy Ef of the fermion in the rest frame of
the annihilating particles,
qjp1j2 + mf2 + jp1j cos
dEf =
jp1jd cos
(C.31)
d vHfF
d vW +fF
Ef =
q
where
= jp3j=
jp3j2 + M W2 and
in the narrowwidth limit,
= (1
2) 1=2. This nally yields the fermion spectrum
vHW BR(W ! f F )
jp1j
F
!HW !HfF ( )
cos = Ef
r
jp1j2+mf2 (C.32)
jp1j
This procedure can be generalized to other threebody nal states in a straightforward
way. For example, for
! W f F , the contribution from the W resonance is
R=W
vW W BR(W
! f F )
jp1j
F
!W W !W +fF ( )
cos = Ef
r
jp1j
jp1j2+mf2
(C.33)
where
F
!W W !W +fF ( ) =
P ; 0; 3 M2!2 M2!32
3 0
2(P ; 3 jM2!32j2)( 13 P
)
;
and M2!32 =
3 (p3) (q)M2!2 is the helicity amlitude for the
! W W annihilation
process. In comparison to before, we have to sum in addition over the polarizations of the
W + that appears in the threebody
nal state. The matrix D 0 ( ) is the same as before.
For swave annihilation the pair of vector bosons is in a state with S = L = 1, J = 0,
and Lz = Sz = 0, when choosing the zaxis along the momentum of one of the nal state
particles. The possible spin projections m1 and m2 of the vector bosons are then determined
by the ClebschGordon coe cients for coupling two spin1 states (S1 = S2 = 1) to a total
spin S = 1 state with m
Sz = 0,
hS1S2; m1m2jS1S2; Smi = <
8 p1=2
m1 = 1; m2 =
m1 = 0; m2 = 0
p1=2 m1 = 1; m2 =
1
1
Since the spatial momenta of the vectors are opposite, this means they can only be in equal
helicity states, and additionally have to be transverse, more precisely
Mp
which implies
(C.34)
(C.35)
(C.36)
(C.37)
F
s wave
!W W !W +fF ( ) =
M W2
3
4
2(M W2
1
m2
1
2
m2
sin2
m2F
2jp1j2 sin2
m2F
This corresponds to the decay spectrum of transversely polasized W bosons, and the last
line applies to massless fermions. It is straightforward to derive the corresponding matrix
D 0 for Z decay and to generalize the procedure to a fermionic (top) resonance. One nds
similarly that due to CP and angular momentum conservation only a de nite helicity state
can contribute, which then determines the decay spectrum.
Open Access.
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