NLO Electroweak Corrections to Higgs Decay to Two Photons

0.12 MB·English

by  Stefano Actis

NLO Electroweak Corrections to Higgs Decay to Two
Photons
Stefano Actis ∗
Institut fu¨r Theoretische Physik E, RWTH Aachen University,
9
D-52056 Aachen - Germany
0
0
2 Therecentcalculationofthenext-to-leadingorderelectroweakcorrectionstothedecay

n of the Standard Model Higgs boson to two photons in the framework of the complex-
a mass schemeis briefly summarized.
J

9
1 Introduction
2

] The production of the Standard Model Higgs boson in photon-photon collisions represents
h an interesting mechanism for measuring the partial width Γ(H → γγ) with a 3% accuracy
p
at an upgrade option of the International Linear Collider (see [2] and references therein).
-
p Therefore,thecomputationofradiativecorrectionstotheleadingorder(LO)decaywidth[3]
e has been an active field of research in the last years.
h
QCD corrections to the partial width of an intermediate-mass Higgs boson have been
[
computedatnext-to-leadingorder(NLO)in[4]andatnext-to-next-to-leadingorder(NNLO)

1 in [5]. The NLO result has been extended in [6] to the entire Higgs-mass range.
v
Electroweak NLO corrections have been evaluated in [7] in the large top-mass scenario
1
8 and in [8] assuming a large Higgs-mass scenario. The two-loop corrections due to light
6 fermions have been derived by the authors of [9], and electroweak effects due to gauge
4 bosons and the top quark have been evaluated through an expansion in the Higgs external
.
1 momentum in [10].
0 More recently, all electroweak corrections at NLO have been computed in [11] for a
9 wide range of the Higgs mass,including the regionacross the W-pair production threshold.
0
It has been shown that the divergent behavior of the amplitude at a two-particle normal
:
v threshold can be removed introducing the complex-mass scheme of [12], a program carried
Xi outin[13]. Notethatwehavenotaddressedthe issueofre-summingCoulombsingularities,
as performed by the authors of [14].
r
a In Section 2 of this note we review the implementation of the complex-mass scheme
in the two-loop computation and the effect on the two-particle threshold region. Next, in
Section3,wediscussthenumericalimpactoftheNLOelectroweakcorrectionsonthepartial
width Γ(H →γγ).
2 Normal thresholds and complex masses
The H →γγ amplitude shows a divergentbehavior around two-particle normal thresholds,
related to the presence of square-root and logarithmic singularities in the variable β2 ≡
i
1−4M2/M2, where M is the Higgs mass and M , with i = W,Z,t, stands for the mass
i H H i
of the W (Z) boson or the top quark.
∗WorksupportedbytheDFGthroughSFB/TR9andbytheECMCRTNHEPTOOLS.Reportnumbers:
PITHA09/02; SFB/CPP-09-06. Slidesavailableat[1].
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Square-root singularities are related to: 1) derivatives of two-point one-loop functions,
associated with Higgs wave-function renormalization; 2) derivatives of three-point one-loop
integrals,generatedbymassrenormalization;3)genuineirreducibletwo-loopdiagramscon-
taining a one-loop self-energy insertion.
Concerning the Higgs wave-function renormalization factor at one loop, it is possible
to explicitly prove that the coefficient of the derivative of the two-point one-loop integral
involvingatopquarkcontainsapositivepowerofthethresholdfactorβ . Therefore,square-
t
root singularities associated with wave-function renormalization appear only at the 2M
W
and 2M thresholds.
Z
We consider now genuine two-
loop diagrams containing a self- W γ
W γ
energy insertion; they naturally H
join terms induced by one-loop W W W W × H W
a W γ b W γ
mass renormalization as shown
t γ t γ
in Figure 1, where bosonic and H t t t × H t
fermionicdiagramsareillustrated. t
Fermionic diagrams of Figure 1c c t γ d t γ
and Figure 1d are β -protected at
t
Figure 1: two-loop (a,c) and mass-renormalization
threshold, and do not require any
(b,d) diagrams relevant for the analysis of square-root
special care. The two-loop irre-
divergencies. Gray circles stand for self-energy inser-
ducible diagram of Figure 1a can
tions, black dots denote derivatives.
be cast in a representation where
the divergent part is completely
written in terms of the one-loop diagrams of Figure 1b. Moreover, it is possible to check
explicitly thattheβ−1 behaviorgeneratedbythe two-loopdiagramexactlycancels the β−1
W W
divergencyduetoone-loopW-massrenormalizationperformedinthecomplex-massscheme.
Therefore, at the amplitude level, square-root singularities are confined to the Higgs-
boson wave-function renormalizationfactor (see also [15]).
As thoroughly dis-
cussed in [11, 13], loga-
W t
γ γ
W t rithmic singularities are
H γ W H γ t generated by the bosonic
W t diagram of Figure 2,
γ γ
W t whereas top-quark terms
areβ -protectedatthresh-
t
Figure 2: irreducible two-loop diagrams which potentially lead old.
to a logarithmic divergency.
A pragmatic gauge-
invariant solution to the
problem of threshold singularities due to unstable particles for the H →γγ decay has been
introduced and formalized in [11]. In this setup (Minimal Complex-Mass scheme) the am-
plitude is written as the sum of divergent and finite terms, and the complex-mass scheme
of [12] is introduced for all gauge-invariant divergent terms. The Minimal Complex-Mass
(MCM) scheme allows for a straightforward removal of unphysical infinities. Real masses
of unstable gauge bosons are traded for complex poles in divergent terms, also at the level
of the couplings, gauge-parameter invariance and Ward identities are preserved and the
amplitude has a decent threshold behavior, as shown in Figure 3 for the NLO electroweak
corrections to the H →γγ decay width (δ ).
EW,MCM
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However,the MCMschemedoes notdealwith
artificial cusps associated with the crossing of WW
4
gauge-boson normal thresholds, as shown in Fig-
2
ure 3 for the WW threshold. In order to cure
these effects, in [13] we have performed a full im- 0
plementation of the complex-mass scheme in the %]-2
d [ dEW,CM
two-loop computation. In the complete Complex- -4 d
EW, MCM
d
Mass (CM) setup, real masses are replaced with -6 EW, real masses
complex poles and the real part of the W-boson -8
self-energy, stemming from mass renormalization
150 152 154 156 158 160 162 164 166 168 170
at one loop, is traded for the full expression, in- Mh [GeV]
cluding its imaginary part. As shown in Figure 3,
Figure 3: percentageNLO electroweak
the full introduction of the CM scheme leads to a
corrections to Γ(H →γγ) in the MCM
complete smoothing of the corrections (δ )
EW,CM
and CM setups described in the text.
at threshold. A more striking effect is associ-
The result obtained using real masses
atedwiththeproductionofaHiggsbosonthrough
(δ )asinputsisalsoshown.
gluon-gluon fusion as discussed in [13]. EW,realmasses
3 Results
Numerical results for the percentage NLO electroweak corrections to the partial width
Γ(H → γγ) are shown in Figure 4. All light-fermion masses have been set to zero in
the collinear-free amplitude and we have defined the W- and Z-boson complex poles by
sj ≡µj (µj −iγj), µ2j ≡Mj2−Γ2j, γj ≡Γj (cid:2)1−Γ2j/(2Mj2)(cid:3),
withj =W,Z. Asinputparametersforthenumericalevaluationwehaveusedthefollowing
values for the masses and the widths of the gauge bosons: M = 80.398GeV, M =
W Z
91.1876GeV, Γ =2.093GeV and Γ =2.4952GeV.
W Z
5
d
d EW, CM
4 d QCD +d
d EW, CM QCD
3 EW, MCM
2
%] 1
[
d
0
-1
-2
-3
100 110 120 130 140 150 160 170
M [GeV]
H
Figure 4: NLO percentage corrections to the H →γγ decay; see text for details.
In Figure 4 we have shown QCD corrections (δ ), electroweak contributions in the
QCD
MCM and CM setups (δ and δ ) and the full NLO prediction involving both
EW,MCM EW,CM
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electroweakeffectsinthecomplex-massschemeandQCDones(δ +δ ). Weobserve
EW,CM QCD
that QCD and electroweak corrections almost compensate below the WW threshold, as
shown by [9, 10], leading to an overall very small correction, well below the expected 3%
accuracyattheplannedlinearcollideroperatingintheγγoption. AbovetheWW threshold,
instead, both corrections are positive and lead to a global 4% effect for M =170 GeV.
H
4 Acknowledgments
G.Passarino,C.SturmandS.Ucciratiaregratefullyacknowledgedforthe collaborationon
the subject. Diagrams have been drawn with Axodraw/Jaxodraw [16].
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