Two-loop top-Yukawa-coupling corrections to the Higgs boson masses in the complex MSSM

0.23 MB·

by  Wolfgang Hollik

MPP–2014–11
Two-loop top-Yukawa-coupling corrections to the
Higgs boson masses in the complex MSSM
4 Wolfgang Hollik and Sebastian Paßehr
1
0
2 Max-Planck-Institut fu¨r Physik

r (Werner-Heisenberg-Institut)
p
F¨ohringer Ring 6, D–80805 Mu¨nchen, Germany
A

3
1

]
h
p
- Resultsfortheleadingtwo-loopcorrectionsof α2 fromtheYukawasectortotheHiggs-boson
p O t
masses of the MSSM with complex parameters are presented. The corresponding self-energies
e (cid:0) (cid:1)
h and their renormalization have been obtained in a Feynman-diagrammatic approach. A numerical
[
analysisofthenewcontributionsisperformedforthemassofthelightestHiggsboson,supplemented

2 by the full one-loop result and the (α α ) terms including complex phases. In the limit of the
t s
v O
real MSSM a previous result is confirmed.
5
7
2
8
.
1
0
4
1
:
v
i
X
r
a
1 Introduction
The discoveryofanew boson[1, 2]withamassaround125.6GeVbythe experimentsATLASand
CMSatCERNhastriggeredanintensiveinvestigationtorevealthenatureofthisparticleasaHiggs
boson from the mechanism of electroweak symmetry breaking. Within the present experimental
uncertainties, which are still considerably large, the measured properties of the new boson are
consistent with the corresponding predictions for the Standard Model Higgs boson [3], but still a
largevarietyofotherinterpretationsispossiblewhichareconnectedtophysicsbeyondtheStandard
Model. Within the theoretical well motivated minimal supersymmetric Standard Model (MSSM),
the observed particle could be classified as a light state within a richer predicted spectrum. The
Higgs sector of the MSSM consists of two complex scalar doublets leading to five physical Higgs
bosons and three (would-be) Goldstone bosons. At the tree-level, the physical states are given by
theneutralCP-evenh,H andCP-oddAbosons,togetherwiththechargedH bosons,andcanbe
±
parametrizedintermsoftheA-bosonmassm andtheratioofthetwovacuumexpectationvalues,
A
tanβ = v /v . In the MSSM with complex parameters, the cMSSM, CP violation is induced in
2 1
the Higgs sector by loop contributions with complex parameters from other SUSY sectors leading
to mixing between h,H and A in the mass eigenstates [4].
Masses and mixings in the neutral sector are sizeably influenced by loop contributions, and
accordingly intensive work has been invested into higher-order calculations of the mass spectrum
from the SUSY parameters, in the case of the real MSSM [5–16] as well as the cMSSM [17–20].
ThelargestloopcontributionsarisefromtheYukawasectorwiththelargetopYukawacouplingh ,
t
or α =h2/(4π), respectively. The class of leading two-loopYukawa-type correctionsof α2 has
t t O t
been calculated so far only in the case of real parameters [11, 12], applying the effective-potential
(cid:0) (cid:1)
method. Together with the full one-loop result [21] and the leading (α α ) terms [20], both
t s
O
accomplished in the Feynman-diagrammatic approach including complex parameters, it has been
implemented inthe public programFeynHiggs[7, 13, 21–23]. Acalculationofthe α2 terms for
O t
the cMSSM, however,has been missing until now.
(cid:0) (cid:1)
In this letter we present this class of α2 contributions extended to the case of complex
O t
parameters. ThecomputationhasbeencarriedoutintheFeynman-diagrammaticapproach;forthe
(cid:0) (cid:1)
specialcaseofrealparametersweobtainaresultequivalenttotheonein[12]inanindependentway,
serving thus as acrosscheck andasa consolidationofformer spectrumcalculationsand associated
tools. These new contributions will be included in the code FeynHiggs.
2 Higgs boson masses in the cMSSM
2.1 Tree-level relations
The two scalar SU(2) doublets can be decomposed according to
= H11 = v1+ √12(φ1−iχ1) , = H21 =eiξ φ+2 , (2.1)
H1 (cid:18)H12(cid:19) −φ−1 ! H2 (cid:18)H22(cid:19) v2+ √12(φ2+iχ2)!
leading to the Higgs potential written as an expansion in terms of the components with the
notation φ−1 = φ+1 †, φ−2 = φ+2 † , h
(cid:0) (cid:1) (cid:0) (cid:1) i
V = T φ T φ T χ T χ
H − φ1 1− φ2 2− χ1 1− χ2 2
φ
1
+ 21 φ1, φ2, χ1, χ2 MM†φφχ MMφχχ!χφ21+ φ−1, φ−2 Mφ±(cid:18)φφ+1+2(cid:19)+... , (2.2)
(cid:0) (cid:1) χ  (cid:0) (cid:1)
 2
 
1
where higher powers in field components have been dropped. The explicit form of the tadpole
coefficients T andofthe massmatricesM canbe foundinRef.[21]. They areparametrizedby the
i
phase ξ, the real SUSY breaking quantities m2 = m˜2 + µ2, and the complex SUSY breaking
1,2 1,2 | |
quantity m2 . The latter can be redefined as real[24] with the help of a Peccei–Quinntransforma-
12
tion [25] leaving only the phase ξ as a source of CP violation at the tree-level. The requirement of
minimizingV atthevacuumexpectationvaluesv andv inducesvanishingtadpolesattreelevel,
H 1 2
which in turn leads to ξ = 0. As a consequence, also M is equal to zero and φ are decoupled
φχ 1,2
from χ1,2 at the tree-level. The remaining 2 2 matrices Mφ,Mχ, Mφ± can be transformed into
×
the mass eigenstate basis with the help of unitary matrices D(x) = −csxx scxx , writing sx ≡ sinx
and c cosx:
x ≡ (cid:0) (cid:1)
h =D(α) φ1 , A =D(β) χ1 , H± =D(β) φ±1 . (2.3)
(cid:18)H(cid:19) (cid:18)φ2(cid:19) (cid:18)G(cid:19) (cid:18)χ2(cid:19) (cid:18)G±(cid:19) (cid:18)φ±2(cid:19)
The Higgs potential in this basis can be expressed as follows,
V = T h T H T A T G
H h H A G
− − − −
h
1 H H+ (2.4)
+ 2 h, H, A, G MhHAGA+ H−, G− MH±G± G+ +....
(cid:18) (cid:19)
(cid:0) (cid:1) G (cid:0) (cid:1)
 
 
with the tadpole coefficients and mass matrices as given in [21]. At lowest order, the tadpoles
vanish and the mass matrices M(0) =diag m2,m2 ,m2,m2 , M(0) =diag m2 ,m2 are
hHAG h H A G H±G± H± G±
diagonal.
(cid:0) (cid:1) (cid:0) (cid:1)
2.2 Mass spectrum beyond lowest order
At higher order, the entries of the Higgs boson mass matrices are shifted according to the self-
energies, yielding in generalmixing of all tree-level mass eigenstates with equal quantum numbers.
In the case of the neutral Higgs bosons the following “mass matrix” is evaluated at the two-loop
level,
M(2) (p2)=M(0) Σˆ(1) (p2) Σˆ(2) (0). (2.5)
hHAG hHAG− hHAG − hHAG
Therein, Σˆ(k) denotes the matrix of the renormalized diagonal and non-diagonal self-energies
hHAG
for the h,H,A,G fields at loop order k. The present approximation for the two-loop part yielding
the leadingcontributionsfromtheYukawasector,treatsthetwo-loopself-energiesatp2 =0forthe
externalmomentum(asdonealsofortheleadingtwo-loop (α α )contributions[20])andneglects
t s
O
contributionsfromthegaugesector(gaugelesslimit). Furthermore,alsotheYukawacouplingofthe
bottomquarkisneglectedbysettingtheb-quarkmasstozero. Thediagrammaticcalculationofthe
self-energies has been performed with FeynArts [27] for the generation of the Feynman diagrams
and TwoCalc [28] for the two-loop tensor reduction and trace evaluation. The renormalization
constants have been obtained with the help of FormCalc [29].
InordertoobtainthephysicalHiggs-bosonmassesfromthedressedpropagatorsintheconsidered
approximation, it is sufficient to derive explicitly the entries of the 3 3 submatrix of Eq. (2.5)
×
corresponding to the (hHA) components. Mixing with the unphysical Goldstone boson yields
subleading two-loop contributions; also Goldstone–Z mixing occurs in principle, which is related
to the other Goldstone mixings by Slavnov-Taylor identities [30, 31] and of subleading type as
well [26]. However, A–G mixing has to be takeninto account in intermediate steps for a consistent
renormalization.
2
The massesof the three neutral Higgs bosonsh ,h ,h , including the new α2 contributions,
1 2 3 O t
are given by the real parts of the poles of the hHA propagator matrix, obtained as the zeroes of
(cid:0) (cid:1)
the determinant of the renormalized two-point vertex function,
detΓˆ p2 =0, Γˆ p2 =i p21 M(2) p2 , (2.6)
hHA hHA − hHA
(cid:0) (cid:1) (cid:0) (cid:1) h (cid:0) (cid:1)i
involving the corresponding 3 3 submatrix of Eq. (2.5).
×
2.3 Two-loop renormalization
For obtainingthe renormalizedself-energies(2.5), countertermshaveto be introducedfor the mass
matrices and tadpoles in Eq. (2.4) up to second order in the loop expansion,
M M(0) + δ(1)M + δ(2)M , (2.7a)
hHAG → hHAG hHAG hHAG
MH±G± → M(H0±)G± + δ(1)MH±G± + δ(2)MH±G±, (2.7b)
T T + δ(1)T + δ(2)T , i=h,H,A,G, (2.7c)
i i i i
→
as well as field renormalizationconstants Z =1+δ(1)Z +δ(2)Z , which are introduced up to
Hi Hi Hi
two-loop order for each of the scalar doublets in Eq. (2.1) through
1 1 1 2
Z = 1+ δ(1)Z + δ(2)Z δ(1)Z . (2.8)
Hi → HiHi 2 Hi 2 Hi − 8 Hi Hi
p (cid:20) (cid:16) (cid:17) (cid:21)
They can be transformedinto (dependent) field-renormalizationconstants for the mass eigenstates
in Eq. (2.3) according to
h Z 0 h h
D(α) H1 D(α)−1 ZhH , (2.9a)
H → 0 Z H ≡ H
(cid:18) (cid:19) (cid:18)p H2(cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
p
A Z 0 A A
D(β) H1 D(β)−1 ZAG , (2.9b)
G → 0 Z G ≡ G
(cid:18) (cid:19) (cid:18)p H2(cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
p
H Z 0 H H
G± →D(β) 0H1 Z D(β)−1 G± ≡ZH±G± G± . (2.9c)
(cid:18) ±(cid:19) (cid:18)p H2(cid:19) (cid:18) ±(cid:19) (cid:18) ±(cid:19)
p
Attheone-looplevel,the expressionsforthe countertermsandforthe renormalizedself-energies
Σˆ(1) (p2) are listed in [21]. For the leading two-loop contributions, we have to evaluate the
hHAG
renormalized two-loop self energies at zero external momentum. In compact matrix notation they
can be written as follows,
Σˆ(2) (0)=Σ(2) (0) δ(2)MZ . (2.10)
hHAG hHAG − hHAG
Thereby, Σ(2) denotesthe unrenormalizedself-energiescorrespondingtothe genuine2-loopdia-
hHAG
gramsanddiagramswithsub-renormalization,asillustratedinFig.1. Thequantitiesinδ(2)MZ
hHAG
can be obtained as the two-loop content of the expression
ZT 0 Z 0
δ(2)MZ = hH M(0) +δ(1)M +δ(2)M hH . (2.11)
hHAG 0 ZT hHAG hHAG hHAG 0 Z
(cid:18) AG(cid:19)(cid:16) (cid:17)(cid:18) AG(cid:19)
Explicit formulae will be given in a forthcoming paper.
The entries of the countertermmatrices in Eq.(2.10) are determined via renormalizationcondi-
tions that are extended from the one-loop level, as specified in [21], to two-loop order:
3
b t t t˜k
Φi tΦ0 t Φj Φt˜ik χ˜2− t˜lΦj Φi˜b1 Φ− t˜kΦj t˜Hk− χ˜n0 H−t˜l Φi Φj H− H−
t t
t˜m t˜l t
Figure 1. Examplesoftwo-loopself-energydiagrams. Thecrossdenotesaone-loopcounterterminsertion.
Φi =h,H,A; Φ0 =h,H,A,G; Φ− =H−,G−.
The tadpole counterterms δ(k)T are fixed by requiring the minimum of the Higgs potential
i
•
not shifted, i.e.1
T(1)+δ(1)T =0, T(2)+δ(2)T +δ(2)TZ =0, i=h,H,A, (2.12a)
i i i i i
with
δ(2)TZ,δ(2)TZ = δ(1)T ,δ(1)T Z , (2.12b)
h H h H hH
(cid:16) (cid:17) (cid:16) (cid:17)
δ(2)TZ,δ(2)TZ = δ(1)T ,δ(1)T Z , (2.12c)
A G A G AG
(cid:16) (cid:17) (cid:16) (cid:17)
whereonlytheone-looppartsoftheZ fromEq.(2.9)areinvolved. T(k) denotetheunrenor-
ij i
malized one-point vertex functions; two-loop diagrams contributing to T(2) are displayed in
i
Fig. 2.
The charged Higgs-boson mass mH± is the only independent mass parameter of the Higgs
•
sector and is used as an input quantity. Accordingly, the corresponding mass counterterm
is fixed by an independent renormalization condition, chosen as on-shell condition, which in
the p2 =0 approximation is given by eΣˆ(k)(0)=0 for the renormalized charged-Higgs self-
ℜ H±
energy, at the two-loop level specified in terms of the unrenormalized charged self-energies
and respective counterterms,
Σˆ(2) = Σˆ(2) , Σˆ(2) (0) = Σ(2) (0) δ(2)MZ , (2.13a)
H± H±G± 11 H±G± H±G± − H±G±
δ(2)MZH±(cid:16)G± = ZTH(cid:17)±G± M(H0±)G± +δ(1)MH±G± +δ(2)MH±G± ZH±G±, (2.13b)
(cid:16) (cid:17)
where only the two-loop content of the last expression is taken. From the on-shell condition,
the independent mass counterterm δ(2)m2H± = δ(2)MH±G± 11 can be extracted.
Thefield-renormalizationconstantsoftheHiggs(cid:0)masseigensta(cid:1)tesinEq.(2.9)arecombinations
•
ofthebasicdoublet-fieldrenormalizationconstantsδ(k)Z andδ(k)Z (k =1,2),whichare
1 2
H H
fixed by the DR conditions for the derivatives of the corresponding self-energies,
δ(k)Z = Σ(k)′(0) div , δ(k)Z = Σ(k)′(0) div . (2.14)
H1 − HH α=0 H2 − hh α=0
h i h i
t tanβ is renormalized in the DR scheme, which has been shown to be a very convenient
β
• ≡
choice [32] (alternative process-dependent definitions and renormalizationof t can be found
β
in Ref. [30]). It has been clarified in Ref. [33, 34] that the following identity applies for the
DR counterterm at one-looporder and within our approximations also at the two-loop level:
1
δ(k)t = t δ(k)Z δ(k)Z . (2.15)
β 2 β H2 − H1
(cid:16) (cid:17)
1Thecounterterms δ(k)TG arenotindependent anddonotneedseparaterenormalizationconditions
4
t˜k t t t˜k t˜k
Φ Φ0 t˜l Φ Φ− b Φ χ˜0n t˜k Φ Φ0 Φ t Φ
t˜m t t t˜l t˜l
Figure 2. Examples of two-loop tadpole diagrams contributing to T(2). The cross denotes a one-loop
i
counterterm insertion. Φi =h,H,A; Φ0 =h,H,A,G; Φ− =H−,G−.
In the on-shell scheme, also the counterterms δM2 M2 and δM2 M2 are required for
• W W Z Z
renormalization of the top Yukawa coupling h = (em ) √2s s M . In the gaugeless
t (cid:14) t β w (cid:14)W
limit these ratios have remaining finite and divergent contributions arising from the Yukawa
(cid:14)(cid:0) (cid:1)
couplings, which have to be included as one-loop quantities h2; they are evaluated from
∼ t
the W and Z self-energies yielding
δM2 Σ (0) δM2 Σ (0) δM2 δM2
W = W , Z = Z , δs2 =c2 Z W . (2.16)
M2 M2 M2 M2 w w M2 − M2
W W Z Z (cid:18) Z W (cid:19)
Inthe Yukawaapproximation,δs2 is finite. The correspondingFeynmangraphsaredepicted
w
in Fig. 4.
As a consequence of applying DR renormalization conditions the result depends explicitly on
the renormalizationscale. Conventionally it is chosen as m being the default value in FeynHiggs.
t
The appearance of δs2 in the α2 terms, as specified above, is a consequence of the on-shell
w O t
scheme where the top-Yukawa coupling h =m /v =m /(vs ) is expressed in terms of
(cid:0) (cid:1) t t 2 t β
1 g e
2
= = . (2.17)
v √2M √2s M
W w W
Accordingly,theone-loopself-energieshavetobeparametrizedintermsofthisrepresentationforh
t
when addedto the two-loopself-energiesinEq.(2.5). Onthe other hand,if the FermiconstantG
F
is used for parametrizationof the one-loop self-energies, the relation
e2
√2G = 1+∆(k)r , (2.18)
F 4s2M2
w W
(cid:16) (cid:17)
has to be applied, whichgets loopcontributionsalso in the gaugelesslimit, atone-loopordergiven
by
c2 δM2 δM2 δs2
∆(1)r = w Z W = w . (2.19)
−s2 M2 − M2 − s2
w (cid:18) Z W (cid:19) w
This finite shift in the one-loop self-energies induces two-loop α2 terms and has to be taken
O t
into account, effectively cancelling all occurrences of δs2.
w (cid:0) (cid:1)
3 Colored-sector input and renormalization
Thetwo-looptopYukawacouplingcontributionstotheself-energiesandtadpolesinvolveinsertions
of counterterms that arise from one-loop renormalization of the top and scalar top t˜ as well as
scalar bottom ˜b sectors. The stop and sbottom mass matrices in the t˜ ,t˜ and ˜b ,˜b bases
L R (cid:0) L(cid:1) R
are given by
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
M = m2q˜L +m2q+MZ2c2β(Tq3−Qqs2w) mq A∗q −µκq , κ = 1 , κ =t , (3.1)
q˜ mq(Aq −µ∗κq) m2q˜R +m(cid:0)2q+MZ2c2β(cid:1)Qqs2w! t tβ b β
5
withQ andT3denotingchargeandisospinofq =t,b. SU(2)invariancerequiresm2 =m2 m2 .
q q t˜L ˜bL ≡ q˜3
In the gaugeless approximationthe D-terms vanish in both the t˜and˜b matrices. Moreover,in our
approximation the b-quark is treated as massless; hence, the off-diagonal entries of the sbottom
matrix are zero and the mass eigenvalues can be read off directly, m2 =m2 =m2 , m2 =m2 .
˜b1 ˜bL q˜3 ˜b2 ˜bR
The stop mass eigenvalues can be obtained by performing a unitary transformation,
Ut˜Mt˜U†t˜=diag m2t˜1,m2t˜2 . (3.2)
(cid:16) (cid:17)
Since A and µ are complex parameters in general, the unitary matrix U consists of one mixing
t t˜
angle θ and one phase ϕ .
t˜ t˜
t˜k,˜b1 t,b t,b t˜m,˜b1 t˜m,˜b1 Φ0,Φ−
t t t t t˜k t˜l t˜k t˜l t˜k t˜l t˜k t˜l
χ˜0,χ˜− Φ0,Φ− χ˜0,χ˜− Φ0,Φ−
n 2 n 2
Figure 3. Feynman diagrams for renormalization of the quark–squark sector. Φ0 =h0,H0,A0,G0;
Φ− =H−,G−.
Five independent parameters are introduced by the quark–squark sector, which enter the two-
loop calculation in addition to those of the previous section: the top mass m , the soft SUSY-
t
breaking parametersm andm (m decouples form =0), and the complex mixing parameter
q˜3 t˜R ˜bR b
At = At eiφAt. On top, µ enters as another free parameter related to the Higgsino sector. These
| |
parameters have to be renormalized at the one-loop level,
m m +δm , M M +δM . (3.3)
t → t t t˜→ t˜ t˜
The independent renormalization conditions for the colored sector are formulated in the following
way:
The mass of the top quark is defined on-shell, i.e.2
•
1
δm = m ˜e ΣL m2 +ΣR m2 +2ΣS m2 , (3.4)
t 2 tℜ t t t t t t
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
according to the Lorentz decomposition of the self-energy of the top quark (Fig. 3)
Σ (p)=pω ΣL(p2)+ pω ΣR(p2)+m ΣS(p2)+m γ ΣPS(p2). (3.5)
t 6 − t 6 + t t t t 5 t
m2 and m2 are traded for m2 and m2 , which are then fixed by on-shell conditions for the
• q˜3 t˜R t˜1 t˜2
top-squarks,
δm2 = ˜eΣ m2 , i=1,2, (3.6)
t˜i ℜ t˜ii t˜i
(cid:16) (cid:17)
involving the diagonal t˜ and t˜ self-energies (diagrammatically visualized in Fig. 3). These
1 2
on-shell conditions determine the diagonal entries of the counterterm matrix
δm2 δm2
Ut˜δMt˜U†t˜= δm2t˜1 δmt˜12t˜2 . (3.7)
t˜1∗t˜2 t˜2 !
2ℜ˜edenotes therealpartofallloopintegrals,butleavesthecouplingsunaffected.
6
The mixing parameter A is correlated with the t˜-mass eigenvalues, t , and µ, through
t β
•
Eq. (3.2). Exploiting Eq. (3.7) and the unitarity of U yields the expression
t˜
µ δµ µ δt
∗ ∗ ∗ β
A δm +m δA + =
(cid:18) t− tβ(cid:19) t t t− tβ t2β !
U U δm2 δm2 +U U δm2 +U U δm2 . (3.8)
t˜11 ∗t˜12 t˜1 − t˜2 t˜21 ∗t˜12 t˜1t˜2 t˜22 ∗t˜11 t˜1∗t˜2
(cid:16) (cid:17)
For the non-diagonal entry of (3.7), the renormalization condition
1
δm2 = ˜e Σ m2 +Σ m2 (3.9)
t˜1t˜2 2ℜ t˜12 t˜1 t˜12 t˜2
h (cid:16) (cid:17) (cid:16) (cid:17)i
is imposed, asin [20], whichinvolvesthe non-diagonalt˜–t˜ self-energy(Fig. 3). By meansof
1 2
Eq.(3.8)thecountertermδA isthendetermined. Actuallythisyieldstwoconditions,for A
t t
| |
and for the phase φ separately. The additionally requiredmass countertermδµ is obtained
At
as described below in section 4.
Asalreadymentioned,therelevantsbottommassisnotanindependentparameter,andhence
•
its counterterm is a derived quantity that can be obtained from Eq. (3.7),
δm2 δm2 =
˜b1 ≡ q˜3
U 2δm2 + U 2δm2 U U δm2 U U δm2 2m δm . (3.10)
| t˜11| t˜1 | t˜12| t˜2 − t˜22 ∗t˜12 t˜1t˜2 − t˜12 ∗t˜22 t˜1∗t˜2 − t t
4 Chargino–neutralino-sector input and renormalization
For the calculation of the α2 contributions to the Higgs boson self-energies and tadpoles, also
O t
the neutralino and chargino sectors have to be considered. Chargino/neutralinovertices and prop-
(cid:0) (cid:1)
agators enter only at the two-loop level and thus do not need renormalization; in the one-loop
terms, however,the Higgsino-massparameterµ appearsandthe countertermδµ is requiredfor the
one-loop subrenormalization. The mass matrices in the bino/wino/higgsino bases are given by
M 0 M s c M s s
1 Z w β Z w β
−
0 M M c c M c s M √2M s
Y = 2 Z w β Z w β and X= 2 W β . (4.1)
M s c M c c 0 µ √2M c µ
− Z w β Z w β − (cid:18) W β (cid:19)
 M s s M c s µ 0 
 Z w β Z w β − 
 
Diagonal matrices with real and positive entries are obtained with the help of unitary matrices
N,U,V by the transformations
N∗YN† =diag mχ˜01,mχ˜02,mχ˜03,mχ˜04 , U∗XV† =diag mχ˜±1 ,mχ˜±2 . (4.2)
(cid:16) (cid:17) (cid:16) (cid:17)
In the gaugeless limit the off-diagonal(2 2) blocks of Y and the off-diagonalentries of X vanish.
×
For this special case the transformation matrices and diagonal entries in Eq. (4.2) simplify,
e2iφM1 0
0
N= 0 e2iφM2 , U= eiφM2 0 , V=1; (4.3a)
1 1 0 eiφµ
 0 1 e2iφµ −  (cid:18) (cid:19)
 √2 i i 
 (cid:18) (cid:19)
 
mχ˜01 =|M1|, mχ˜02 =|M2|, mχ˜03 =|µ|, mχ˜04 =|µ|, mχ˜±1 =|M2|, mχ˜±2 =|µ|; (4.3b)
and only the Higgsinos χ˜03, χ˜04, χ˜±2 remain in the O α2t contributions.
(cid:0) (cid:1)
7
The Higgsino mass parameter µ is an independent input quantity and has to be renormalized
accordingly, µ µ+δµ, fixing the counterterm δµ by an independent renormalization condition,
→
which renders the one-loopsubrenormalizationcomplete. Together with the soft-breaking parame-
tersM andM , µcanbedefinedintheneutralino/charginosectorbyrequiringon-shellconditions
1 2
for the two charginosandone neutralino. However,since only δµ is requiredhere,it is sufficient to
impose a renormalization condition for χ˜±2 only; the appropriate on-shell condition reads,
1
δµ= eiφµ µ e ΣL (µ2)+ΣR (µ2) +2 e ΣS (µ2) , (4.4)
2 | |ℜ χ˜±2 | | χ˜±2 | | ℜ χ˜±2 | |
n h i h io
where the Lorentz decomposition of the self-energy for the Higgsino-like chargino χ˜±2 (see Fig. 4)
has been applied, in analogy to Eq. (3.5).
t˜k,˜b1 t˜k t t˜k,˜b1 t,b t˜k,˜b1
χ˜± χ˜± W W W W Z Z Z Z W,Z W,Z
2 2
b,t ˜b1 b t˜l,˜b1 t,b
Figure 4. Feynman diagrams for thecounterterms δµ, δMW/MW, and δMZ/MZ.
Another option implemented in the α2 result is the DR renormalizationof µ, which defines
O t
the countertermδµin the DRscheme, i.e. bythe divergentpartofthe expressioninEq.(4.4). For
(cid:0) (cid:1)
the numerical analysis and comparison with [12] the DR scheme is chosen at the scale m .
t
5 Numerical analysis
In this section we focus on the lightest Higgs-boson mass derived from Eq. (2.6) and present re-
sults for m in different parameter scenarios. In each case the complete one-loop results and the
h1
(α α )termsareobtainedfromFeynHiggs,whilethe α2 termsarecomputedbymeansofthe
O t s O t
correspondingtwo-loopself-energiesspecifiedintheprevioussectionswiththeparametersµ,t and
(cid:0) (cid:1) β
the Higgs field-renormalization constants defined in the DR scheme at the scale m . Thereby, the
t
new α2 self-energiesarecombinedwith the results ofthe otheravailableself-energiesaccording
O t
to Eq. (2.5) within FeynHiggs, and the masses are derived via Eq. (2.6). For comparison with
(cid:0) (cid:1)
previous results G is chosen for normalization as mentioned at the end of section 2.3.
F
Theinputparametersforthenumericalresultsinthissectionareshowninthefiguresorthecap-
tions, respectively,when they arevaried. The residualparameters,being the sameforallthe plots,
are chosen as M =200 GeV, M = 5s2 3c2 M , and m =m =m =m =2000 GeV
2 1 w w 2 ˜lL q˜L ˜lR q˜R
for the first two sfermion generations, together with the Standard Model input m =173.2 GeV
(cid:0) (cid:1)(cid:14)(cid:0) (cid:1) t
and α =0.118.
s
Asafirstapplication,westudythecaseoftherealMSSM,whereananalyticresultofthe α2
O t
contributions is known [12] from a calculation making use of the effective-potential method. The
(cid:0) (cid:1)
versionof FeynHiggsfor realparametershas this result included, makingthus a directcomparison
with the prediction of our new diagrammatic calculation possible. Thereby all parameters and
the renormalization have been adapted to agree with Ref. [12]. Very good agreement is found
between the two results that have been obtained in completely independent ways. As an example,
this feature is displayed in Fig. 5, where the shift from the α2 terms in the two approaches
O t
are shown on top of the mass prediction without those terms. The grey band depicts the mass
(cid:0) (cid:1)
range 125.6 1 GeV around the Higgs signal measured by ATLAS and CMS. The mass shifts
±
8
effectivepotential Feynmandiagrams withoutOHΑ2L
t
125
120 tΒ=30
Μ=200GeV
m
h1 115 mŽ =mŽ =mŽ =1000GeV
@GeVD q3 tR bR
110 mŽ{3=mΤŽR=1000GeV
mŽ =1500GeV
g
m =800GeV
105 A0
A =A =A
t b Τ
100
-2000 -1000 0 1000 2000
A @GeVD
t
Figure 5. Comparison of the result for the lightest Higgs-boson mass in the effective potential approach
(blue)andtheFeynman-diagrammaticapproach(red). Thecurvesarelyingontopofeachother,indicating
the agreement of both calculations in the limit of real parameters. For reference the result without the
contributions of O(cid:0)α2t(cid:1) is shown (yellow). The grey area depicts the mass range between 124.6 GeV
and 126.6 GeV.
displayed in Fig. 5 underline the importance of the two-loop Yukawa contributions for a reliable
prediction of the lightest Higgs boson mass.
InthepresentversionofFeynHiggsforcomplexparameters,thedependenceofthe α2 terms
O t
on the phases of φ and φ is approximated by an interpolation between the real results for the
At µ (cid:0) (cid:1)
phases 0 and π [23, 35]. A comparison with the full diagrammatic calculation yields deviations
±
thatcanbe notable,inparticularforlarge A . Fig. 6displaysthe qualityofthe interpolationasa
t
| |
functionofφ andshowsthatthedeviationsbecomemorepronouncedwithrisingµ,whichiskept
At
real. [Also the admixture of the CP odd part in h is increasing with µ, but it is in general small,
1
below2%.] Theasymmetricbehaviourwithrespecttoφ iscausedbythephaseofthegluinomass
At
in the (α α ) contributions. The shaded area again illustrates the interval [124.6,126.6]GeV.
t s
O
In Fig. 7 the dependence onµ is shownfor the massshift originatingfromthe α2 terms and
O t
for the full result for the lightest Higgs-boson mass, choosing different values for the phase φ .
(cid:0) (cid:1) At
Particularly for large µ the results can vary significantly and lead to different predictions for the
lightest Higgs mass. The kinks around µ 1200 GeV and µ 1450 GeV arise from physical
≈ ≈
thresholds of the decay of a stop into a higgsino and top.
6 Conclusions
We have presented new results for the two-loop Yukawa contributions α2 from the top–stop
O t
sector in the calculation of the Higgs-boson masses of the MSSM with complex parameters. They
(cid:0) (cid:1)
generalize the previously known result for the real MSSM to the case of complex phases entering
at the two-loop level; in the limit of real parameters they confirm the previous result. Combining
the new terms with the existing one-loop result and leading two-loop terms of (α α ) yields
t s
O
an improved prediction for the Higgs-boson mass spectrum also for complex parameters that is
equivalent in accuracy to that of the real MSSM. In the numerical discussion we have focused on
the mass of the lightest neutralboson, m , which receives specialinterest by comparisonwith the
h1
mass ofthe recentlydiscoveredHiggssignal. The massshifts originatingfromthe α2 termsare
O t
(cid:0) (cid:1)
9