One-loop corrections, uncertainties, and approximations in neutralino annihilations

13 Pages·2011·0.35 MB·English

PHYSICAL REVIEW D 84, 116001 (2011)
One-loop corrections,uncertainties, and approximations in neutralino annihilations: Examples
F. Boudjema,1 G. Drieu La Rochelle,1,2 and S. Kulkarni3
1LAPTh*, Universite´ deSavoie, CNRS, B.P.110, Annecy-le-Vieux F-74941,France
2CERN Physics Department, Theory Division, CH-1211 Geneva 23, Switzerland
3Bethe Center for Theoretical Physics and Physikalisches Institut, Universita¨t Bonn, Nussallee 12, D-53115 Bonn, Germany
(Received23 August 2011; published 8 December 2011)
Theextractedvalueoftherelicdensityhasreachedthefewpercentlevelprecision.Onecantherefore
no longer content oneself with calculations of this observable where the annihilation processes are
computed at tree level, especially in supersymmetry where radiative corrections are usually large.
Implementing full one-loop corrections to all annihilation processes that would be needed in a scan
over parameters isa daunting task. On the other hand one may ask whether most of the corrections are
takenintoaccountthrougheffectivecouplingsoftheneutralinothatimprovethetree-levelcalculationand
wouldbeeasy toimplement.Weaddressthisissuebyconcentratinginthisfirststudyontheneutralino
couplingto(i)fermionsandsfermionsand(ii)Z.Afterconstructingtheeffectivecouplingswecompare
theirefficiencytothefullone-loopcalculationandcommentonthefailuresandsuccessoftheapproach.
As a bonus we point out that large nondecoupling effects of heavy sfermions could in principle be
measured intheannihilationprocess,apointofinterestinviewof thelatestlimitonthe squarkmasses
from the LHC. We also comment on the scheme dependencies of the one-loop corrected results.
DOI: 10.1103/PhysRevD.84.116001 PACS numbers: 95.35.+d, 11.10.Gh, 11.30.Pb, 98.80.(cid:1)k
practicallyallanalysesthataimatconstrainingtheparame-
I. INTRODUCTION
ter space of the minimal supersymmetric standard model
With barely 1 fb(cid:1)1 of data, the LHC is pushing many (MSSM) through the relic density are based on tree-level
hithertopopular,thoughnaive,extensionsofsupersymme- cross sections of the annihilation processes entering the
try to the corners of high masses [1] while leaving some predictions of this quantity which, as stressed, is experi-
hope for a discovery of a rather light Higgs particle that mentally given within the percent accuracy. Only seldom
could still be compatible with supersymmetry [2]. Before do some analyses assign a theory uncertainty to these
this very recent paradigm, supersymmetric models (and annihilation cross sections, an uncertainty due essentially
most models of new physics for that matter) were very to the fact that higher order loop effects are not known.
strenuouslyconstrainedtoathinsliverinparameterspace, This uncertainty, in the rare case where it is taken into
most notably from the very precise measurement of the account, is however assumed to be invariably the same
dark matter relic density that has now reached the few whateverthenatureofthedominantprocessandthecom-
percent level and that will get even more precise in the position of the LSP. The reason the loop corrections are
future, hence cornering even further model building. ignored, irrespective of the model specified, is that the
Combining the results of the 7-year WMAP data [3] on calculation of the relic density requires most often the
the 6-parameter (cid:1)CDM model, the baryon acoustic oscil- evaluation of a large number of processes. Most analyses
lationsfromSDSS[4]andthemostrecentdeterminationof are done with public codes [7–9] based on tree-level cal-
the Hubble constant [5] one [6] arrives at (cid:2)CDMh2 ¼ culations. Computations of the relic density at one loop
0:1123(cid:2)0:0035, where (cid:2)CDM is the density of cold dark have now been achieved for quite a few channels [10–14]
matter (CDM) normalized to the critical density, and h is andtoolsexistnowtoperforminprincipleanycalculation
the Hubble constant in units of 100 kms(cid:1)1Mpc(cid:1)1. One oftherelicdensitybeyondtree-levelamplitudesthanksto
hasreachedaprecisionof3%.ThedatafromtheLHCdo the recent development of adapted automation tools [15].
notinferthatdarkmatterwithinsupersymmetry,exempli- Beside the findings [10–14] that these corrections are
fiedmostnicelythroughtheneutralino,thelightestsuper- important, the improvements have not percolated into
symmetricparticle(LSP),isoforderof1TeVorso;thisis most analyses. It must be said that these calculations do
justalimitonthecoloredconstituentsofthemodel.Asfor
involvesomenontrivialissuesabouttherenormalizationof
the Higgs particle, were it not for the large radiative
the MSSM and more generally techniques for one-loop
correctionsonthemassofthelightestmember,supersym-
integrals thatcertainly requireexpertise. The other reason
metry wouldlongbe a forlorn construct. The Higgs parti-
isthatevenwhentheycouldbeimplemented,theyarestill
cle is the most prominent example where radiative
certainly extremely CPU time consuming, forbidding any
corrections are far from negligible in supersymmetry, yet
attempts of fits, likelihood search, and in a more general
contextanysamplingoftheparameterspace,especiallyif
*UMR 5108 du CNRS,associe´e a` l’Universite´ de Savoie. onetakesintoaccountthefactthattheMSSMismorethan
1550-7998=2011=84(11)=116001(13) 116001-1 (cid:1)2011 American Physical Society
F. BOUDJEMA, G. DRIEU LA ROCHELLE, AND S. KULKARNI PHYSICAL REVIEW D 84, 116001 (2011)
liberal with unconstrained parameters. Yet, apart from case. One might argue that the bino case corresponds to
providingamorepreciseprediction,one-loopcalculations what was referred to as the bulk region in the constrained
can probe higher masses, a situation akin to the precision MSSMandislargelyruledout,whereasforaHiggsinothis
electroweakobservablesandtheirsensitivitytothetopand would not be the dominant process. As we have just
Higgsmass.Forexample,nondecouplingcorrectionshave stressed, the aim is not to strive to find a good scenario.
been revealed in one-loop calculations of supersymmetry Besides, it suffices to change the cosmological ingredients
observables [16–18]. They stem from the superoblique [21]enteringthecalculationoftherelicdensitytorevivethe
contributionwhichistermedinanalogytothemorefamil- so-calledbulkregion.Wewillseethatbyconsideringthese
iarobliquecorrectionsthatappearinone-loopcorrections few simple examples the conclusions about the efficacy of
to 4-fermion processes in the standard model. They origi- the effective couplingis quite different. Moreover, there is
nate from the universal, process-independent self-energy noneedheretoconvertafullcorrectedcrosssectionintoa
corrections. An example inview of the recent findings of relic density value; we rather take for all the models we
theLHCisthatsuperheavysquarksleaveanon-negligible study the annihilation cross section for an energy corre-
imprintonmanyobservables,inparticular,theannihilation sponding to a relative velocity of v¼0:2, typical for the
cross sections involvingdark matter. relicdensitycalculation.Asknown,forzerorelativeveloc-
The aim of this paper is two-fold. First, to stress again itytheprocessenjoyschiralitysuppressionwhichisliftedat
theimportanceoftheloopcorrectionsfortherelicdensity higher order through gauge boson radiation; however the
and show again that even when a loop calculation is effectontherelicistotallymarginal[10].
available, there still remains in some cases an uncertainty The paper is organized as follows. We first briefly de-
that pertains to the choice of the renormalization scheme. scribe the ingredients necessary to perform a one-loop
Thesecondandmoredetailedaspectistodiscusswhether calculation in supersymmetry covering both automation
anapproximationtotheone-loopcalculationcanbefound. and renormalization schemes; it is in this section that we
We aim at implementing a universal correction through will write down the effective universal neutralino cou-
effective couplings of the LSP and checking its validity plings as well as some definitions. Section III contains
against a complete one-loop calculation. If such an ap- our main results. After a few definitions we first study
proximation is possible and general enough, it could be the case of a binolike neutralino before addressing the
implemented in existing codes (based on tree-level cross caseofaHiggsino-likeLSP.Wealsoquantifythepossible
sections) calculating the relic density. Such was the case uncertaintiesduetotheschemedependence.Weconclude
withtheinclusionoftheSommerfeldeffect[19]inthecase inSec. IV by somegeneral observations.
of coannihilation or processes dominated by Higgs ex- Throughout the paper we use some shorthand notation
change for which (cid:3)mb [20] corrections are included [7]. for angles. Generically c(cid:3), s(cid:3), t(cid:3) stand for cos(cid:3), sin(cid:3),
This preliminary study takes a simple process, namely tan(cid:3). The weak mixing angle (cid:3)W is defined as cos(cid:3)W ¼
(cid:1)~01(cid:1)~01 !(cid:2)þ(cid:2)(cid:1),asatestingground.Theaimofthisstudy cW ¼MW=MZ, where MW is the W mass and MZ the Z
isnottofindagoodscenariothatreturnsthecorrectactual mass.
valueoftherelicdensitybuttotrytounravelsomegeneral
common features of the loop calculation to improve the
II. CALCULATIONS,RENORMALIZATION
predictionsoftherelicdensity.Theaimistoratherfindout
SCHEMES AT FULL ONE-LOOP
whether one can improve on the tree-level calculation by
introducing effective couplings of the LSP that could be A. Tree-level considerations
used for any process. As we will see, though at first sight At tree level (see Fig. 1), (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1) proceeds
1 1
naive,theprocess(cid:1)~01(cid:1)~01 !(cid:2)þ(cid:2)(cid:1)embodiesthethreetypes through (i) t-channel smuon exchange dominated by a
of couplings of the neutralino: to f=f~, gauge bosons, and (cid:2)~ inthecaseofabinolikesinceithasthelargesthyper-
R
Higgs particles. Here we cover the bino and the Higgsino charge; (ii) a Z exchange which, on the other hand, is
FIG. 1. Tree-level diagrams contributing to (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1). (a) is the t-channel (cid:2)~ exchange, (b) is the Z exchange, and (c) an
1 1
exampleof a Higgs exchange.
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ONE-LOOP CORRECTIONS, UNCERTAINTIES, AND ... PHYSICAL REVIEW D 84, 116001 (2011)
suppressedforthebino;and(iii)Higgsexchange,butthis mindsinceonewouldbetemptedtouseaDRschemefor
is small in view of the Yukawa coupling of the muon. (cid:4), defined as M , to take into account the fact that dark
Z
Therefore, as advertised, all types of couplings for the matter is annihilating at roughly the electroweak scale, so
LSParepresent:tofermionsinthe(cid:1)~01(cid:2)~(cid:2)coupling,gauge that (cid:4)ðð2m(cid:1)~0Þ2Þ is a more appropriate choice. One should
bosons in the (cid:1)~01(cid:1)~01Z, and Higgs scalars such as (cid:1)~01(cid:1)~01A0 rememberth1attheuseof(cid:4)ðð2m(cid:1)~0Þ2Þinsteadoftheon-shell
(A0isthepseudoscalarHiggs).Itisthroughthechoiceofa 1
value in the Thomson limit would correct the tree-level
hierarchy in the set M1, M2, (cid:2) that we can largely define cross section for (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1) by about 14%, indepen-
1 1
the nature of the LSP. Numerically speaking we call a dently of the neutralino mass in the range 50<M(cid:1)~0 <
neutralino pure or almost pure when its mixing to the 1
300 GeV.Aswewillsee and havereportedelsewherefor
specified species is over 99%.
other processes this running does not, most of the time,
takeintoaccount thebulkoftheradiativecorrectionsthat
B. Renormalization and loop corrections, general
we report here. Therefore for further reference, let us
considerations and issues
introducethe correction due to the running of (cid:4),
1. Set upof the automatic calculation: SLOOPS (cid:3) ¼(cid:5)(cid:4)eff (cid:1)(cid:5)0 ¼2(cid:3)(cid:4); (1)
One-loop processes calculated via the diagrammatic (cid:4) (cid:5)0
Feynman approach involve a huge number of diagrams
wherethecrosssection(cid:5)0 isthetreelevelcalculatedwith
evenfor2!2reactions,especiallyinatheorylikesuper- (cid:4)0 ¼(cid:4)ð0Þ¼1=137:0359895, whereas (cid:5)(cid:4)eff is the tree
symmetry.Performingafullone-loopcalculationbyhand level with (cid:4)0 !(cid:4)eff:ðQ2Þ¼(cid:4)ðQ2Þ¼(cid:4)0=ð1þ(cid:3)(cid:4)ðQ2ÞÞ,
wfuiltlhoounte-aluotoopmcaatilocnulaistiopnracisticdaollnyeuwntirtahcttahbeleh.eOlpuroefxtahcet Q¼2m(cid:1)~01.Withourinputparameters(cid:3)(cid:4)ðMZ2Þ¼0:06.In
therunningweallowforallchargedparticlesincludingthe
automated code SLOOPS. SLOOPS is an automated code
W boson contribution, the top and the sfermions, and the
forone-loopcalculationsinsupersymmetry.Itisacombi-
charginos, though for the light LSP scenario we consider
nation of LANHEP [22], the bundle FEYNARTS [23], theseadded contributions arevery small1
FORMCALC [24], and an adapted version of LOOPTOOLS (ii)TheHiggssector:thepseudoscalarHiggsmassM0 is
[25,26]. LANHEP deals with one of the main difficulties used as an input parameter while insisting on vanishAing
thathastobetackledfortheautomationoftheimplemen-
tadpoles. t , which at tree level is the ratio of the two
(cid:6)
tationofthemodelfile,whichisenteringthethousandsof
expectation values of the Higgs doublets, can be defined
verticesthatdefinetheFeynmanrules.Onthetheoryside,
throughseveral schemes:
aproperrenormalizationschemeneedstobesetup,which
(a) In the Dabelstein-Chankowski-Pokorski-Rosiek
then means extending many of these rules to include
(DCPR) scheme [31,32] t is defined by requiring
counterterms. This part is done through LANHEP which that the (renormalized) A(cid:6)0Z transition vanishes at
allowsonetoshiftfieldsandparametersandthusgenerate Q2 ¼M2 .
counterterms most efficiently. The ghost Lagrangian is A0
(b) A DR definition where the t counter term, (cid:7)t , is
derived directly from the Becchi-Rouet-Stora-Tyutin (cid:6) (cid:6)
defined as a pure divergence leaving out all finite
charge (BRST) transformations. The loop libraries used
parts.
in SLOOPS are based on LOOPTOOLS with the addition of
(c) A process-dependent definition of this counterterm
quite a few routines, in particular, those for dealing with
byextractingitfromthedecayA0 !(cid:8)þ(cid:8)(cid:1) thatwe
small Gram determinants that appear in our case at small
will refer to as A for short. This definition is a
relativevelocitiesoftheannihilatingdarkmatter,andeven (cid:8)(cid:8)
good choice for the gauge independence of the
more so of relevance forindirect detection [26].
processes.
(d) An on-shell definition with the help of the mass of
2.Renormalization
the heavy CP Higgs H taken as input parameter
In SLOOPS all sectors of the MSSM are implemented called the MH scheme from now on. We have
through a one-loop renormalization. This is explained in reported elsewhere that this scheme usually intro-
detail in [10,13,27,28]. Here we only briefly sketch the duces large radiative corrections.
renormalization procedure. We have worked, as far as
possible, within an on-shell scheme generalizing what is These schemes are critically reviewed in [27]. By default
done for the electroweak standard model [29]. weusetheDCPRschemebutwhenquantifyingtheeffect
(i) The standard model (SM) parameters: the fermion of the scheme dependence on t we also use the DR and
(cid:6)
masses as well as the mass of the W and Z are taken as MH scheme.
inputphysicalparameters.Theelectricchargeisdefinedin
theThomsonlimit;seeforexample[29].Thelightquarks
1FortheW bosoncontributiontheself-energyofthephotonis
(effective) masses are chosen [30] so as to reproduce the calculatedinanonlineargauge[29]correspondingtotheback-
SM value of (cid:4)(cid:1)1ðMZ2Þ¼127:77. This should be kept in ground gauge in order to maintain Uð1ÞQED gauge invariance.
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F. BOUDJEMA, G. DRIEU LA ROCHELLE, AND S. KULKARNI PHYSICAL REVIEW D 84, 116001 (2011)
(iii)Thesfermionsector:fortheprocessathandonlythe (cid:1)0 ¼Ncn: (2)
smuonparametersrequirerenormalization.Fortheslepton
sector we use as input parameters masses of the two N diagonalizesthemassmixingmatrixY intheneutralino
charged sleptons which in the case of no-mixing define sector;see[28]fordetailsandconventions.Althoughonly
theR-sleptonsoftbreakingmass,M(cid:2)~R andtheSUð2Þmass, (cid:1)~01entersourcalculationswedoneedtofixalltheelements
M(cid:2)~ ,givingacorrectiontothesneutrinomassatoneloop. that define its composition and hence couplings. For this
L
Thoughnotneededhere,inthesquarksectoreachgenera- sectorweimplementanon-shellschemebytakingasinput
tionneedsthreephysicalmassestoconstrainthebreaking threemassesinordertoreconstructtheunderlyingparame-
parameterMQ~L fortheSUð2Þpart,Mu~R,Md~R fortheRpart. ters M1, M2, (cid:2). In SLOOPS [28] the default scheme is to
See [28] fordetails. choose two chargino masses m(cid:1)~(cid:2) and m(cid:1)~(cid:2) as input to
1 2
(iv) The chargino/neutralino sector: first of all, for the define M2 and (cid:2) and one neutralino mass to fix M1. The
neutralinosattreelevelthephysicalfields(cid:1)0,i¼1;...;4 massesoftheremainingthreeneutralinosreceiveone-loop
i
are obtained from the current eigenstates ðcnÞt ¼ quantum corrections. In this scheme, these counterterms
ðB~0;W~0;H~0;H~0Þ through a unitary complex matrix N are [28]
1 2
(cid:1) (cid:2)
(cid:7)M ¼ 1 ðM m2 (cid:1)(cid:2)detXÞ(cid:7)m(cid:1)~þ1 þðM m2 (cid:1)(cid:2)detXÞ(cid:7)m(cid:1)~þ2 (cid:1)M2 ðM þ(cid:2)s Þ(cid:7)MW2 (cid:1)(cid:2)M2 s c (cid:7)t(cid:6) ;
2 M22(cid:1)(cid:2)2(cid:1) 2 (cid:1)~þ1 m(cid:1)~þ1 2 (cid:1)~þ2 m(cid:1)~þ2 W 2 2(cid:6) MW2 W 2(cid:6) 2(cid:6) t(cid:6) (cid:2)
(cid:7)(cid:2)¼ 1 ð(cid:2)m2 (cid:1)M detXÞ(cid:7)m(cid:1)~þ1 þð(cid:2)m2 (cid:1)M detXÞ(cid:7)m(cid:1)~þ2 (cid:1)M2 ð(cid:2)þM s Þ(cid:7)MW2 (cid:1)M M2 s c (cid:7)t(cid:6) ;
(cid:2)2(cid:1)M22 (cid:1)~þ1 2 m(cid:1)~þ1 (cid:1)~þ2 2 m(cid:1)~þ2 W 2 2(cid:6) MW2 2 W 2(cid:6) 2(cid:6) t(cid:6)
(3)
1
(cid:7)M1 ¼N12ið(cid:7)m(cid:1)0i (cid:1)N22i(cid:7)M2þ2N3iN4i(cid:7)(cid:2)(cid:1)2N1iN3i(cid:7)Y13(cid:1)2N2iN3i(cid:7)Y23(cid:1)2N1iN4i(cid:7)Y14(cid:1)2N2iN4i(cid:7)Y24Þ; (4)
wofitihthdenteXut¼ralMino2(cid:2)de(cid:1)finMedW2 se2n(cid:6)tir[e2l8y].f(cid:7)romm(cid:1)~0iitisssthelef-ceonuenrgteyr;tesreme (cid:1)~0i !(cid:1)~0i þ12Xð(cid:7)ZijPLþ(cid:7)Z(cid:4)ijPRÞ(cid:1)~0j: (5)
j
[28]. (cid:7)O represents the shift on the parameter O that gen-
erates the counterterm for that quantity. Looking at these
equations some remarks can be made. First, in the special
configuration M2(cid:3)(cid:2)(cid:2) an apparent singularity might (vi) Dimensional reduction is used as implemented in
arise. Reference [10] pinpointed this configuration which the FEYNARTS-FORMCALC-LOOPTOOLS bundle at one-loop
can induce a large t -scheme dependence in the counter- through the equivalent constrained dimensional renormal-
(cid:6)
terms(cid:7)M1;2 and(cid:7)(cid:2).Suchamixedscenarioisnotcovered ization [33].
here. Second, the choice of m(cid:1)~0 as an input parameter is
1
appropriate only if the lightest neutralino is mostly bino 3.Infrared divergences
(jN11j(cid:3)1) or if the binolike neutralino is not too heavy For the processes (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1), we can decompose
compared to other neutralinos. Indeed we can see that if the one-loop amplitude1s1in a virtual part MEW and a
aNn1di(cid:3)ma0ytihnetrcooduuncteeratelramrge(cid:7)Mfin1itiescsourbrjeeccttioton;lathrgiseiusnrceelarttaeidnttyo counterterm(CT)contributionMCT.Thesumo1flotohpesetwo
amplitudes must be ultraviolet finite and gauge indepen-
the fact that M1 is badly reconstructed. To avoid such
dent. Because of the virtual exchange of the massless
uncertaintyweonlychooseiasthemostbinolike;inother
wordsinEq.(4),jN1ij¼MaxðjN1jjÞ;j¼1;...4. photon, this sum can contain infrared divergencies. This
iscuredbyaddingasmallmasstothephotonand/orgluon,
v) Finally diagonal field renormalization is fixed by
(cid:9) and (cid:9) . This mass regulator should exactly cancel
demanding the residue at the pole of the propagator of all (cid:10) g
against the one present in the final-state radiation of a
physicalparticlestobeunity,andthenondiagonalpartby
photon. The QED contribution is therefore split into two
demanding no mixing between the different physical par-
parts:asoftonewherethephotonenergyE isintegrated
ticleswhenonshell.Thisisimplementedinallthesectors. (cid:10)
Inourcaseapartfromthemuon,thisstepisimportantfor to less than some small cutoff kc and a hard part with
the(cid:1)~01.WeinsistthatNij isusedbothatthetreeleveland E(cid:10)>kc. The former requires a photon mass regulator.
one-looplevel.Nonethelesstodefinethephysicalstatewe Finally the sum M1loopþMCTþMs(cid:10)oftðE(cid:10)<kcÞþ
dointroducetheshiftfortheneutralinos[28]throughwave MhardðE >k Þshouldbeultravioletfinite,gaugeinvari-
(cid:10);g (cid:10);g c
function renormalization ant, not depend on the mass regulator,and on the cut k .
c
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ONE-LOOP CORRECTIONS, UNCERTAINTIES, AND ... PHYSICAL REVIEW D 84, 116001 (2011)
4.Checking theresult renormalization of the muon is specific to the muon final
state.Theboxesthefour-pointone-particleirreducible(1-PI)
(i)Foreachprocessandsetofparameters,wefirstcheck
functions,aswellastheQEDcorrectionarealsospecificto
the ultravioletfinitenessof the results. Thistestappliesto
theprocess.Theconstructoftheuniversalcorrectionforthe
thewholesetofvirtualone-loopdiagrams.Theultraviolet
finiteness test is performed by varying the ultraviolet pa- effectivecoupling(cid:1)~01ff~from(cid:1)~01(cid:2)(cid:2)~ isdifferentfromthatof
rameterCUV ¼1=","istheusualregulatorindimensional (cid:1)~01(cid:1)~01Z, since in the latter all three particles can be consid-
reduction.WevaryCUV by7ordersofmagnitudewithno ered as universal. For example the full correction to the
change in the result. We content ourselves with double vertex (cid:1)~01(cid:2)(cid:2)~ shown in Fig. 2(a) consists of a 1-PI three-
precision. point function vertex correction (triangle) which is muon
(ii) The test on the infrared finiteness is performed by specific and that does not need to be calculated to buildup
including both the loop and the soft bremsstrahlung con- theeffectivecoupling.Italsocontainswavefunctionrenor-
tributionsandcheckingthatthereisnodependenceonthe malizationoftheneutralinosaswellascountertermsforthe
fictitious photonmass (cid:9) . gaugecouplingsandforotheruniversalquantitiessuchast
(cid:10) (cid:6)
(iii) Gauge parameter independence of the results is whichmustbecombinedtoarriveattheuniversalcorrection
essential. It is performed through a set of the eight gauge for the (cid:1)~0ff~ vertex. The aim of the paper is therefore to
1
fixing parameters based on the implementation of a non- extract these process-independent contributions and define
lineargauge [27]. effectiveverticesfortheLSPinteractions.Thisisakintothe
effective coupling of the Z to fermions where universal
C. Effective couplings forneutralino correctionsaredefined.Describingthebulkoftheradiative
interactions vsfull calculation corrections in terms of effective couplings has been quite
successfultodescribe,forexample,theobservablesattheZ
1. Contributions at full one loop
peak. Although not describing most perfectly the effect of
The full set of one-loop contributions to the process (cid:4)
the full corrections for all observables (for example Zbb
(cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1) consists of two-point functions (self-
1 1 receives an important triangle contribution due to the large
energies and transitions such as (cid:1)~0 !(cid:1)~0), vertex three-
1 2 topYukawacoupling)onemustadmitthattheapproachhas
pointfunctionsasinFigs.2(a)and2(b),andboxdiagrams.
donequiteagoodjob.Mostoftheeffectivecorrectionswere
The vertex corrections include also the counterterms; the
universal, described in terms of a small set of two-point
latterasexplainedpreviouslyinvolvetwo-pointfunctions.
functionsofthegaugebosons.
Tothese,oneshouldalsoaddtheQEDfinal-stateradiation.
The other benefit was that such approximations were
sensitivetonondecouplingeffectsthatprobehigherscales
2.Effective couplings of theneutralino atone loop
(topmassandHiggsmass).Thesetoftwo-pointfunctions,
Among this full set of corrections one can construct a andfor(cid:1)~0(cid:1)~0Zthree-pointfunctions,shouldofcourselead
1 1
finitesubsetthatisnotspecifictothemuon.Thissubsetwill to a finite and gauge-invariant quantity. Loops involving
be involved in all processes involving neutralinos. For ex- gauge bosons have always been problematic (even in the
ample,thevertexcorrectionto(cid:1)~0(cid:1)~0Zisobviouslyindepen- caseoftheZff(cid:4))insuchanapproachsinceitisdifficultto
1 1
dentofthemuonbeinginthefinalstate;asimilarstatement extractagauge-independentvalue.Forthecouplingsofthe
can be said for (cid:1)~0(cid:1)~0h=H=A. Also, all occurrences of the neutralinos as would be needed for approximating their
1 1
wave function renormalization of the neutralino (including annihilation cross section independently of the final state,
transitionsbetweenneutralinos)andtheZareprocessinde- one would therefore expect that apart from the rescaling
pendent.Thesamecanbesaidalsoofthecountertermstothe of the gauge couplings which can be considered as an
gauge couplings and the vacuum expectation values or in overall constant, the mixing effect between the different
other words v, t . On the other hand the wave function neutralinosshouldbeaffected.Onecaninfactreorganizea
(cid:6)
FIG. 2. Different types of corrections appearing at one-loop for the process (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1). (a) is the correction tovertex (cid:1)~0(cid:2)(cid:2)~
1 1 1
forthet-channel(cid:2)~ exchange,(b)isthefullcorrectiontothe(cid:1)~0(cid:1)~0Zvertexforthes-channelZexchange,and(c)isanexampleofa
1 1
box loop.
116001-5
F. BOUDJEMA, G. DRIEU LA ROCHELLE, AND S. KULKARNI PHYSICAL REVIEW D 84, 116001 (2011)
fewofthetwo-pointfunctions(thatcanbewrittenalsoas [27,28]. (cid:7)g=g¼(cid:7)e=e(cid:1)(cid:7)s =s , (cid:7)g0=g0 ¼(cid:7)e=e(cid:1)
W W
counterterms) to define an effective coupling for the neu- (cid:7)c =c , and (cid:7)s =s ¼c2(cid:7)t =t . Equation (7) agrees
W W (cid:6) (cid:6) (cid:6) (cid:6) (cid:6)
tralino. One should of course also correct in this manner with what was suggested in [34]. Let us stress again that
the Z(cid:2)þ(cid:2)(cid:1) coupling. Let us stress again that in this first in these self-energies no gauge bosons and therefore no
investigationwewillprimarilytakeintoaccounttheeffects neutralinos and charginos are taken into account but just
of fermions and sfermions in the universal loops. For the sfermionsandfermions;otherwisethiswouldnotbefinite.
(cid:1)~01(cid:1)~01Z effective coupling we also attempt to include the For binolike case, self-energies containing gauge and
virtual contribution of the gauge bosons especially since Higgs bosons (with their supersymmetric counterparts)
fortheHiggsino-likecasethecouplingstoWandZarenot arenotexpectedtocontributemuch.Thisisnotnecessarily
suppressed. the case for winos and Higgsinos.
3. The effective (cid:1)~01ff~ 4.The effective (cid:1)~01(cid:1)~01Z
Tofindtheprocess-independentcorrectionstothiscou- Since all particles making this vertex can now be con-
pling, we recall that in the basis ðB~0;W~0;H~01;H~02Þ before sidered as being process independent (as far as neutralino
mixing and for both f the couplings for the two chiral annihilations are concerned), all counterterms including
L;R
Lorentz structures writeas wave function renormalization of both the Z and (cid:1)~0 must
1
(cid:1) (cid:2)
1 1 gm gm be considered. The price to pay now is that the genuine
pffi2ffiffiðg0Yf;g(cid:8)3f;y1;f;y2;fÞ¼pffi2ffiffi g0Yf;g(cid:8)3f;M cu ;M sd trianglevertexcorrections(cid:1)~01(cid:1)~01Zmustalsobeincluded.It
(cid:1) W (cid:6)(cid:2) W (cid:6) is only the sum of the vertex and the self-energies that
! g0;g; g ; g : (6) renders a finite result. When correcting this vertex one
M c M s must also correct the Z(cid:2)þ(cid:2)(cid:1) vertex keeping within the
W (cid:6) W (cid:6)
spirit of calculating the universal corrections. This can be
Yf, (cid:8)f3 are the isospin and SUð2Þ charges of the corre- implemented solely through self-energy corrections (ex-
sponding fermion/sfermions. The two Higgsinos couple cluding the muon self-energies) and there is no need to
differently to the up and down fermions with a coupling calculateherethegenuinevertexcorrections.Anexception
thatisproportionaltotheYukawacoupling.Thoughthisis wouldbetheproductionofthebandtosomeextentthetop
not universal we can still isolate a universal part where quark where genuine vertex corrections are important.
thereisnoreferencetothefinalfermion/sfermion.Thisis Talking of heavy flavors, when computing the correction
what is meant by the last expression in Eq. (6), where the tothe(cid:1)~0(cid:1)~0ZwiththeZoffshellwithaninvariantmassQ2,
1 1
explicit mass of the corresponding fermion masses has oneshouldalsoincludethe(cid:1)~0(cid:1)~0G0vertex,whereG0isthe
1 1
been dropped. The variations/counterterms on these pa- neutralGoldstone boson.In our casewe restrict ourselves
rameters have to be implemented before turning to the to almost massless fermions. The case of the top and
physical basis. The latter as explained in the previous bottom final states will be addressed elsewhere together
paragraph is achieved through the diagonalizing matrix N withthepotentialrelevantcontributionoftheHiggsparti-
[Eq. (2)] as in tree-level supplemented by wave function cles in the s-channel.
renormalizationwhichinvolvebothdiagonalandnondiag- Since one is including the genuine 1-PI vertex correc-
onaltransitionsoftheneutralino;seeEq.(5).Inthecaseof tion, it is important to inquire whether this correction
effectivecouplingofneutralinos,thisisachievedbydefin- generates a new Lorentz structure beyond the one found
inganeffectivemixingmatrixsuchthatN !Nþ(cid:3)N(cid:1)ff~ at tree level. The contribution to the tree-level Lorentz
in all couplings of the neutralino.The (cid:3)N(cid:1)ff~write as structure is finite after adding the self-energies and the
vertex. Any new Lorentz structure will on the other hand
(cid:3)N(cid:1)ff~¼(cid:7)g0N þ1XN (cid:7)Z ; be finite on its own. General arguments based on the
i1 g0 i1 2 j1 ji Majorananatureoftheneutralinosbackedbyournumeri-
j
~ (cid:7)g 1X calstudiesshowthatnonewLorentzstructureisgenerated
(cid:3)N(cid:1)ff ¼ N þ N (cid:7)Z ; for neutralinos. First of all, at tree level one has only one
i2 g i2 2 j2 ji
j structure
(cid:1) (cid:2)
(cid:3)Ni(cid:1)3ff~¼ (cid:7)gg(cid:1)12(cid:7)MM2W2 (cid:1)(cid:7)cc(cid:6) Ni3þ12XNj3(cid:7)Zji; (7) L(cid:1)~01(cid:1)~01Z ¼g4ZðN13N13(cid:1)N14N14Þ(cid:1)~01(cid:10)(cid:2)(cid:10)5(cid:1)~01Z(cid:2);
(cid:1) W (cid:6) (cid:2) j e (8)
(cid:3)Ni(cid:1)4ff~¼ (cid:7)gg(cid:1)12(cid:7)MM2W2 (cid:1)(cid:7)ss(cid:6) Ni4þ12XNj4(cid:7)Zji; gZ ¼cWsW:
W (cid:6) j
The overall strength is a consequence of the fact that the
where j runs from 1to 4 and for the LSP,i¼1. coupling emerges solely from the Higgsino with a gauge
All the counterterms above are calculated from self- coupling.IndeedintheðB~0;W~0;H~0;H~0Þbasisthecoupling
1 2
energy two-point functions and are fully defined in is /gZð0;0;1;(cid:1)1Þ. Only the Lorentz structure (cid:10)(cid:2)(cid:10)5
116001-6
ONE-LOOP CORRECTIONS, UNCERTAINTIES, AND ... PHYSICAL REVIEW D 84, 116001 (2011)
survivesasaconsequenceoftheMajorananature.Withp1, Eq. (11) which is a nontrivial result. Moreover this con-
p2 denoting the incoming momenta of the two (cid:1)~01, at tribution is gauge parameter independent in the class of
one-loop a contribution (p(cid:2)(cid:1)p(cid:2)) does not survive sym- (linear) and nonlinear gauge fixing conditions [29]. To
1 2
metrization, whereas (p(cid:2)þp(cid:2)) will not contribute for weigh up the gauge/gaugino/Higgsino contribution, we
1 2
massless muons. We calculate this correction for a Z willthereforealsocomparewiththisgeneralizedeffective
with an invariant mass Q2; in the application this Q2 will geff including all virtual particles. Observe that in
(cid:1)~0(cid:1)~0Z
1 1
be set to the invariant mass of the muon pair. This vertex Eq. (11) we have the contribution (cid:5) ð0Þ which vanishes
contributionisdenoted(cid:3)g4 ðQ2Þ(cid:5)(cid:3)g4 .Thecon- (cid:10)Z
(cid:1)~0(cid:1)~0Z (cid:1)~0(cid:1)~0Z for fermions and sfermions but which is essential for the
1 1 1 1
tributionofthecouplingcountertermsdefiningg andthe contributionofthevirtualW.Inanycase,includinggauge
Z
Z wave function renormalization define the universal cor- bosonsintherenormalizationofelectromagneticcoupling
rection to the Z coupling strength geff ¼g ð1þ(cid:3)g Þ, requiresthe inclusionof the (cid:5) ð0ÞinEq. (11) forgauge
Z Z Z (cid:10)Z
with (cid:3)g =g ¼(cid:7)g =g þ(cid:7)Z =2. (cid:7)Z is the wave invariance to be maintained [29]. We stress that we will
Z Z Z Z ZZ ZZ
function renormalization of the Z. We of course have to present the effect of the generalized effective coupling
addthewavefunctionrenormalizationofthe(cid:1)~0 likewhat geff as an indication of the gauge boson contribution
was done for the (cid:1)~01ff~vertex. We improve on1this imple- w(cid:1)~h01i(cid:1)~l01eZkeepinginmindthatthisresultmayleadtounitarity
mentation by taking into account the fact that the Z is off violation.Indeed,throughcuttingrules,theW loopcanbe
shell and therefore the wave function renormalization seen as made up of the scattering WþW(cid:1) !Z!(cid:2)þ(cid:2)(cid:1)
through (cid:7)ZZZ ¼(cid:5)0ZZðMZ2Þ is only part of the correction thatneedsacompensationfromthecutintheboxshownin
that would emerge from the correction to the complete Z Fig. 2(c). For the effective Z(cid:2)þ(cid:2)(cid:1) coupling we only
propagator in the s-channel contribution with invariant includethefermion/sfermioncontributioninEqs.(11)and
mass Q2. Note that here there is no need for including a (12); adding the gauge bosons would require part of the
Z(cid:10) transition since photons do not couple to neutralinos. 1-PI triangle contribution to Z!(cid:2)þ(cid:2)(cid:1).
Collecting all these contributions, the effective vertex is
obtained by making gZ !ge(cid:1)~f0f(cid:1)~0Z and Ni1 !Ni1þ III.ANALYSIS
1 1
(cid:3)N(cid:1)~01(cid:1)~01Z with Sincewewill be studyingdifferent compositionsofthe
i1
neutralinoswewilltakedifferentvaluesforthesetM1,M2,
geff ¼g ð1þ(cid:3)g ðQ2Þþ(cid:3)g(cid:3) ðQ2ÞÞ; (9)
(cid:1)~0(cid:1)~0Z Z Z (cid:1)~0(cid:1)~0Z (cid:2).On the other hand, the default parameters in the Higgs
1 1 1 1
sector are
1X
(cid:3)Ni(cid:1)~j01(cid:1)~01Z ¼2 Nkj(cid:7)Zki;ði;j;kÞ¼1...4: (10) MA0 ¼1 TeV t(cid:6) ¼4: (13)
k
Thesfermionsectorisspecifiedbyaratherheavyspectrum
Explicitly (inparticularwithinthelimitssetbytheLHCforsquarks
(cid:1) (cid:2) [1]). All sleptons left and right of all generations have a
1 s (cid:5) ð0Þ
(cid:3)g ¼ (cid:5)0 ð0Þ(cid:1)2 W (cid:10)Z common mass which we take to be different from the
Z 2 (cid:1)(cid:10)(cid:10) (cid:2)(cid:1)cW MZ2 (cid:2) commonmassinthesquarksector.Alltrilinearparameters
þ1 1(cid:1)c2W (cid:5)ZZðMZ2Þ(cid:1)(cid:5)WWðMW2 Þ Af (includingthoseforstopsandsbottom)aresetto0.The
2 s2 M2 M2 default values for the sfermion masses are
W Z W
(cid:1) (cid:2)
1 (cid:5) ðQ2Þ(cid:1)(cid:5) ðM2Þ M~ ¼M~ ¼500 GeV;
(cid:1) ZZ ZZ Z : (11) lR lL
2 Q2(cid:1)MZ2 Mu~R ¼Md~R ¼MQ~L ¼800 GeV; (14)
Atthesametimeforthefermionwithchargeq wecorrect A ¼0:
theZff(cid:4) /gZð(cid:10)5þð1(cid:1)4jqfjs2WÞÞ(cid:10)(cid:2) byeffecftivelymak- f
ing g !g ð1þ(cid:3)g Þ with (cid:3)g defined in Eq. (11) and By default we will focus on relatively light neutralinos
s2 toZ Z Z Z (around 100 GeV) scattering with a relative velocity
W v¼0:2.
(cid:1) (cid:2)
c2 (cid:5) ðM2Þ (cid:5) ðM2 Þ c (cid:5) ðk2Þ To analyze consistently the efficiency of effective cor-
(cid:3)s2 ¼ W ZZ Z (cid:1) WW W þ W (cid:10)Z : (12)
W s2 M2 M2 s k2 rections wewill refer to the following quantities:
W Z W W
(cid:5) (cid:1)(cid:5)
By default we include only the fermions and sfermions in (cid:3)eff ¼ eff 0: (15)
(cid:5)
thevirtualcorrectionsdescribedbyEqs.(11)and(12).For 0
the(cid:1)~01(cid:1)~01Zoneexpectthecontributionofthegaugebosons Here (cid:5)eff is thecrosssectioncalculated withtheeffective
and the neutralinos/charginos to be non-negligible espe- couplings that include, by default, universal process-
cially for the Higgsino case. In fact, including such con- independent particles excluding gauge bosons and gaugi-
tributionsstillgivesanultravioletfiniteresultforgeff in nos/Higgsinos. We will explicitly specify when including
(cid:1)~0(cid:1)~0Z
1 1
116001-7
F. BOUDJEMA, G. DRIEU LA ROCHELLE, AND S. KULKARNI PHYSICAL REVIEW D 84, 116001 (2011)
20
20
15 15
%) Full One-Loop %) Full One-Loop
Effective Effective
∆ ( 10 Noneffective ∆ ( 10 Noneffective
5 5
0 0
50 100 150 200 250 300 350 0 5 10 15 20 25 30 35 40
M1 (GeV) tβ
FIG. 3(coloronline). Correctionstothetree-levelcrosssectionfortheprocess(cid:1)~01(cid:1)~01 !(cid:2)þ(cid:2)(cid:1)inthebinocaseasafunctionofM1
(leftpanel)andt (rightpanel).Weshowthefullone-loop,theeffectivecorrection,andthedifferencewhichwetermnoneffective.
(cid:6)
M2 ¼500, (cid:2)¼(cid:1)600 GeV.
all virtual particles in those corrections, referring to it as be admitted though that the bulk of the correction is
(cid:3)W. This correction will be compared to the correction through the running of (cid:4). To see how general this con-
eff
solelyduetotherunningoftheelectromagneticcoupling; clusion is we scanned over the set ðM1;M2;(cid:2)Þ while
see Eq. (1). To see how well the correction through the maintaining (cid:1)~0 with a 99% binolike component. This is
1
effective couplings (cid:1)~01ff~ and (cid:1)~01(cid:1)~01Z reproduces the full simply obtained by taking M2 ¼500, (cid:2)¼(cid:1)600 GeV,
one-loop correction we introduce and scanning up to M1 ¼350 GeV. We also checked
(cid:3)NE¼(cid:5)one-lo(cid:5)op(cid:1)(cid:5)eff; (cid:3)full¼(cid:5)one-lo(cid:5)op(cid:1)(cid:5)0; (16) ht(cid:6)owfrsoemnsi2tivtoeo4u0r.cTohneclussuipoenrsisymdempetenrdyi-nbgreoanktin(cid:6)gbysfvearrmyiinogn
0 0
masses were first left to their default values. As Fig. 3
with (cid:5)one-loop the full one-loop cross section, (cid:3)NE mea- shows, our conclusions remain quantitatively unchanged.
sures what we will refer to as the noneffective (NE) cor- Thereisnoappreciabledependenceint ,wearriveatthe
(cid:6)
rections since this measures the remainder of all the same numbers as our default t value. As for the depen-
(cid:6)
correctionsthatarenottakenintoaccountbytheeffective dence in M1, it is very slight; for M1(cid:3)50 GeV there is
vertices approach. (cid:3)full ¼(cid:3)eff þ(cid:3)NE is the full one-loop perfect matching with our effective coupling implementa-
correction. tion, then as M1 increases to 350 GeV, the nonuniversal
corrections remain negligible, below 2%.
A. Bino case Theannihilationofneutralinosandhencetherelicden-
sityisaverygoodexampleofthenondecouplingeffectsof
1. Effective vs full corrections
veryheavysparticles,aremnantofsupersymmetrybreak-
We first take ðM1;M2;(cid:2)Þ¼ð90;200;(cid:1)600Þ GeV, ing. The variation in the fermion/sfermion masses is all
which yields a lightest binolike neutralino (the bino com-
contained in the effective couplings that we have intro-
position is 99%) with mass m(cid:1)~0 ¼91 GeV. At tree level duced.Leavingthedependenceonthesmuonmassattree
1
thecrosssectionforrelativevelocityv¼0:2is(cid:5)b~ ¼ level, and the very small (see below) contribution of the
6:75(cid:6)10(cid:1)3 pb. Note for further reference that th(cid:2)isþ(cid:2)is(cid:1)an smuon to the 1-PI vertex (cid:1)~01(cid:2)(cid:2)~, the bulk of the smuon
massdependenceiswithintheeffectivecoupling.Figure4
order of magnitude larger than annihilation into a pair of
W’s: (cid:5)b~ ¼4:51(cid:6)10(cid:1)4 pb. The annihilation pro-
WþW(cid:1)
ceedspredominantlythroughthet-channel;binoscoupling
20
toZareverymuchreduced.Thisleadstothefollowingset
of corrections: 15
%) Full One-Loop
(cid:3)eff ¼17:52%ð(cid:3)(cid:4) ¼14:56%Þ (17) ∆ ( 10 NonEeffffeeccttiivvee
(cid:3) ¼2:06%ð(cid:3) ¼19:58%Þ:
NE full 5
For our first try, the effective universal coupling does 0
400 800 2000 3000
remarkablywell,fallingshortofonly2%correctioncom-
~
Mq (GeV)
pared to the full calculation. Note that although the most
naiveimplementationthrougharunningoftheelectromag-
FIG. 4 (color online). Corrections to the tree-level cross sec-
neticcouplingfaresalsoquitewellitisnonetheless5%off tionfortheprocess(cid:1)~01(cid:1)~01 !(cid:2)þ(cid:2)(cid:1) inthebinocase(M1 ¼90,
the total correction, therefore the effective correction M2 ¼200, (cid:2)¼(cid:1)600 GeV) as a function of the common soft
through the effective couplings performs better. It must supersymmetry-breaking squark mass.
116001-8
ONE-LOOP CORRECTIONS, UNCERTAINTIES, AND ... PHYSICAL REVIEW D 84, 116001 (2011)
TABLE I. Coefficients of the lnðmfÞ (running couplings) lnðmf~Þ (nondecoupling effects) in
(cid:3)eff. ðc;sÞ givevery similar results to ðu;dÞ.
aQ(cid:4)L au(cid:4)R ad(cid:4)R af af
e 0.0010 0.00231 0.00310 0.15%
ðu;dÞ 0.000575 0.00236 0.000698 (0.00413, 0.00103) 0.15%
ðt;bÞ (cid:1)0:00406 0.00838 0.000661 (0.00413, 0.00103) 0.15%
shows how the correction increases as the mass of the 19:58%ðDCPRÞ; 19:79%ðDRÞ; 19:51%ðMHÞ:
squarks increases from 400 GeV to 3 TeV; we take here
a common mass for the supersymmetry-breaking squark
This confirms that the t scheme dependence is very
(cid:6)
masses (both right and left in all three generations). The
negligible. For the bino case, it is natural to reconstruct
nonuniversal correction of about 2% is insensitive to this
changeinsquarkmasseswhereasboth(cid:3)eff and(cid:3)NE show dMe1pefnrodmenctheeonLeScPh;onoosneestahneolethsseranneaulytrzailnignot,hseayM(cid:1)~10s,cwhheimche
2
thesamelogarithmgrowththatbringsa3%changeasthe
in our example is winolike. This introduces more uncer-
squarkmassisvariedintherange400GeVto3TeV.This
taintyorerrorsincewiththisschemethecorrectionsattain
result also confirms that genuine vertex corrections and
24.08%, more than 4% compared tothe usual scheme.
boxcorrectionsareverysmall.Wehavealsoextractedthe
individual contribution of each species of fermions to the
B. Higgsino case
totalnondecouplingeffectofsfermions.Toachievethiswe
numerically extracted the logarithm dependence of the 1. Effective versus full corrections
nondecoupling effect for each species of sfermions. We
Inthebinocaseourtrialpointhadaneutralinoofmass
have parametrized the effective correction as 91 GeV. We therefore take the point (600, 500, (cid:1)100)
(cid:3)f ¼af~lnmf~=Q(cid:1)aflnmf=Qþbf with wHhigicghsingoivceosnteantL.STPhewsfiethrmMio(cid:1)n~01p¼ara9m5eGteerVsarweitthheadef9a9u%lt
(18)
Q¼2m(cid:1)~0 f~¼d~Rþu~RþQ~L: values.IntheHiggsinocasethecrosssectionisdominated
1 bytheexchangeoftheZinthes-channel,sothebulkofthe
correctionsthroughtheeffectivecouplingswillbethrough
ThecoefficientsofthefitaregiveninTableI.Asexpected
the effective (cid:1)~0(cid:1)~0Z. For further reference note that the
thefittoa isextremelywellreproducedbytherunningof 1 1
f
b(cid:4), i¼.e,ba.fT¼heNfictqt2fo34a(cid:11)(cid:4).iWsmeaadlesotofivnadlidaae~te¼thae(cid:8)~fi¼tparo(cid:2)~c,ebduer¼e. t(cid:5)reh(cid:2)~eþ-(cid:2)le(cid:1)v¼el 2c:r5o8ss(cid:6)s1e0c(cid:1)ti3onpbf,ortinaynnaihnidlattiootnalliyntoinsmigunoifincsanist
T(cid:2)he mo(cid:8)st importantfobservation is that the stops behave especially compared to annihilation into W, (cid:5)hW~þW(cid:1) ¼
differently; this is due to the Yukawa coupling of the top 18:83 pb. This is an observation we will keep in mind.
andmixing.Iftherewerenotacompensationbetweenthe The one-loop corrections we find for (cid:5)h~ are
left and right contribution of the stops (compare to u~) the (cid:2)þ(cid:2)(cid:1)
contribution of the stops would be even more important
ðfor(cid:2)¼(cid:1)100GeVÞ
and would dominate. Considering the different contribu-
tionsandthescalesthatenterourcalculations,itisdifficult (cid:3) ¼13:55ð(cid:3) ¼14:62%Þ (19)
eff (cid:4)
to attempt giving an analytical result. Leaving the stop
(cid:3) ¼(cid:1)21:09%ð(cid:3) ¼(cid:1)7:54%Þ:
aside, the different contributions to a~ can be roughly NE full
f
approximated by y2N N =8=c2 , N ¼2 for doublets and
f c d W d Thisresultisinaquitestrikingcontrasttothebinocase.
1forasingletofSUð2Þ.y isthehypercharge,correspond-
f The effective coupling does not reproduce at all the full
ing to the couplings of the sfermions to the bino
correction and is off by as much as 21%. It looks like, at
component.
least for this particular choice of parameters, that going
through the trouble of implementing the effective (cid:1)~0(cid:1)~0Z
1 1
2. Scheme dependence in the bino case was in vain since this correction is, within a percent,
Wehavecomparedthefullcorrectiontoanapproximate reproduced by the naive running of (cid:4). As we will see,
effective implementation and observed that the approxi- bothoftheseconclusionsdepend muchontheparameters
mation is quite good. However, even the full correction, of the Higgsino and even the squark masses. For example
being computed at one loop, is potentially dependent on consider (cid:2)¼(cid:1)50 GeV, leaving all other parameters the
the renormalization scheme chosen. As discussed earlier, same.Ofcoursethisisapurelyacademicexercise,sincein
weanalyzethet(cid:6) schemedependenceandtheM1 scheme this case, the charginos with mass m(cid:1)(cid:2)1 ¼55 GeV are
dependence. For t we obtain the following corrections: ruledout by LEP data. Nonetheless, inthis case
(cid:6)
116001-9
F. BOUDJEMA, G. DRIEU LA ROCHELLE, AND S. KULKARNI PHYSICAL REVIEW D 84, 116001 (2011)
20
ðfor(cid:2)¼(cid:1)50 GeVÞ (cid:3)eff ¼10:7ð(cid:3)(cid:4) ¼12%Þ (20) 10
(cid:3) ¼(cid:1)6:9%ð(cid:3) ¼3:8%Þ: 0
NE full
-10
Had we included all particles in the effective vertex, we %) -20
would get a correction (cid:3)Weff ¼4:4%, improving thus the ∆ ( -30 Full One-Loop
agreement with the one-loop correction for this particular -40 Effective
value of (cid:2) up to 0.6%. At the same time a correction in -50 Boxes
Effective (All)
terms of a running of (cid:4) will be off by more than 8%. -60
-200 -180 -160 -140 -120 -100 -80 -60 -40
These two examples show that one cannot, in the
µ (GeV)
Higgsinocase,drawageneralconclusionontheefficiency
oftheeffectivecouplingaswasdoneinthebinocase.Let
FIG. 5 (color online). Corrections to the tree-level cross sec-
usthereforelookathowthecorrectionschangewith(cid:2),and tion for the process (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1) in the Higgsino case as a
1 1
therefore with the mass of the LSP, while maintaining its function of (cid:2). Shown are the effective vertex correction
Higgsino nature. We have varied (cid:2) from (cid:1)200 GeV to (Effective, with only fermions/sfermions in the loops), the ef-
(cid:1)40 GeV. Figure 5 shows that the full correction is ex- fective(cid:1)~01(cid:1)~01Zcouplingincludingallparticles[denotedEffective
tremely sensitive to the value of (cid:2). For (cid:2)¼(cid:1)200 GeV (All)], the non-QED boxes (Boxes), and the full correction.
the full one-loop correction is as much as (cid:1)42%, casting M2 ¼500, M1 ¼600 GeV.
doubt on the loop expansion. The effective coupling cor-
rectionswithonlyfermions/sfermionsontheotherhandis our study. Figure 7 shows how the effective (with only
much smoother and positive, bringing about 10% correc- fermions and sfermions) and the full correction get modi-
tion. Including all particles in the effective (cid:1)~01(cid:1)~01Z vertex fiedwhenthecommonmassofallsquarks(allgenerations,
brings in an almost constant reduction of about 6%. leftandright)increasesfrom400GeVto3TeV.Tobetter
Therefore as thevalue of j(cid:2)j increases, the effective one- illustrate the important effect of the Yukawa coupling of
loop corrections in the case of the Higgsino cannot be the top/stop sector, we plot the corrections also for m ¼
t
trusted. The same figure shows that the behavior and the 0:1 GeV. For m ¼170:9 GeV, the correction drops by
t
increase in the corrections are due essentially to the con- about 13% when the mass of the squarks increases from
tributionofthe boxes. Herethe boxes meanthenon-QED 400GeVto3TeV.Thisismuchmoredramaticthaninthe
box (involving an exchange of a photon which is infrared bino case where we observed a 3% increase in the same
divergentbeforeincludingtherealphotonemission2).The range. Observe that for our default squark mass of
large contribution of the boxes can be understood by 800 GeV, the effective correction including sfermions/
looking at the box in Fig. 1(c). Indeed, as argued previ- fermions is such that it almost accidentally coincides
ously, cutting through the box reveals that it represents with the running of (cid:4). If one switches off the top quark
(cid:1)~01(cid:1)~01 !WþW(cid:1) production that rescatter into (cid:2)þ(cid:2)(cid:1). mass,insteadofa13%decreaseweobservean8%increase
Both these processes have very large cross sections com- form ¼0:1 GeV!Observethatthedifferenceoneseesfor
pared to the tree-level (cid:1)~01(cid:1)~01 !(cid:2)þ(cid:2)(cid:1). Our conclusion is mQ~ ¼t 400 GeV between mt ¼170:9 GeV and mt ¼
therefore that the effectivevertex approximation is inade- 0:1 GeV is due essentially to the running of (cid:4) with very
quate as soon as the channel (cid:1)~0(cid:1)~0 !WþW(cid:1) opens up.
1 1 light top that accountsfor 3%.
When this occurs, in practical calculations of the relic
density, the channel (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1) is irrelevant and
1 1
must rather analyze the loop corrections to (cid:1)~01(cid:1)~01 ! 15
WþW(cid:1). This process was studied in [10,13] and will be 10
Full One-Loop
investigated further through an effective approximation 5 Effective
inaforthcomingstudy.Ontheotherhand,thedependence 0 Noneffective
tohfouthgeh rtehlearteiveisccoerrrteacintiloynmoonret(cid:6)deipsenqdueitnecemtohdaenstinevtehne ∆ (%) -1-50
binocase,especiallyatlowervaluesoftan(cid:6).Thisisshown -15
in Fig.6. -20
We now investigate the nondecoupling of very heavy -25
5 10 15 20 25 30 35 40
squarks (and heavy sfermions in general). Since we are
inaHiggsinoscenario,weexpecttheYukawacouplingof tβ
the fermions to play a more prominent role than what
FIG. 6 (color online). Corrections to the tree-level cross sec-
was observed in the bino case. This is well supported by tion for the process (cid:1)~0(cid:1)~0 !(cid:2)þ(cid:2)(cid:1) in the Higgsino case as a
1 1
function of t . We show the full one-loop, the effectivecorrec-
(cid:6)
2The contribution of the QED box þ real photon emission is tion, and the remainder (Noneffective). (cid:2)¼(cid:1)100, M2 ¼500,
only 0.1% (cid:2)¼(cid:1)600 GeV.
116001-10