Resonance--Continuum Interference in Light Higgs Boson Production at a Photon Collider

0.18 MB·English

by  Lance J. Dixon

SLAC–PUB–13498
Resonance–Continuum Interference in Light Higgs Boson
Production at a Photon Collider
9
0
0 Lance J. Dixon and Yorgos Sofianatos
2
SLAC National Accelerator Laboratory
n
a
J Stanford University, Stanford, CA 94309, USA

7
(Dated: December, 2008)

]
h
Abstract
p
-
p We study the effect of interference between the Standard Model Higgs boson resonance and the
e
h continuum background in the process γγ H b¯b at a photon collider. Taking into account
[
→ →

2 virtual gluon exchange between the final-state quarks, we calculate the leading corrections to the
v
2 height of the resonance for the case of a light (m < 160 GeV) Higgs boson. We find that the
H
1
7
interference is destructive and around 0.1–0.2% of the peak height, depending on the mass of the
3
.
2 Higgs and the scattering angle. This suppression is smaller by an order of magnitude than the
1
8 anticipated experimental accuracy at aphoton collider. However, thefractional suppressioncan be
0
:
v significantly larger if the Higgs coupling to b quarks is increased by physics beyond the Standard
i
X
Model.
r
a
PACS numbers: 14.80.Bn, 13.66.Fg,14.80.Cp
1
The Standard Model of Particle Physics (SM) has been very successful in describing a
wide range of elementary particles phenomena to high accuracy. A key ingredient of the
model is the scalar Higgs field, responsible for electroweak symmetry breaking and for gen-
erating the masses of essentially all massive elementary particles [1, 2, 3]. Similar fields exist
in extensions of the SM, such as the Minimal Supersymmetric Standard Model (MSSM). In
the SM, the Higgs boson is the only particle that remains undiscovered, and its properties
aredetermined byitsmass. It isamaingoalofcurrent andfuturehighenergy physics exper-
iments to identify the Higgs boson and explore the details of the Higgs sector. In particular,
the discovery of the Higgs boson could take place at Run II of the Tevatron at Fermilab;
if not there, then at the Large Hadron Collider (LHC) at CERN. Precise measurements of
its properties will be one of the tasks of the proposed International Linear Collider (ILC).
There is an option to use the ILC as a photon collider, by backscattering laser light off of
the high energy electron beams. The high energy, highly polarized photons produced in this
way can be used to study the various Higgs couplings to very high accuracy [4, 5, 6, 7, 8, 9].
The mass of the Higgs boson in the SM and MSSM has already been constrained by
experiment to a range well within the reach of the aforementioned designed machines. Pre-
cision electroweak measurements have put an upper bound on the allowed values for its
mass, m . 170 GeV at 95% confidence level in the SM [10, 11]. In the MSSM the Higgs
H
boson mass obeys the bound m m at tree level; radiative corrections increase this limit
H Z
≤
to about 135 GeV [12, 13, 14]. The mass of the Higgs boson has also been bounded from
below via the Higgs-strahlung process e+e HZ at LEP2, with m & 114.1 GeV in the
− H
→
SM and m & 91.0 GeV in the MSSM [15, 16, 17, 18, 19, 20].
H
At a photon collider, among the two possible modes, γγ and eγ, the former is especially
useful for Higgs physics. For m < 140 GeV, the most important channel involves Higgs
H
¯
production via photon fusion, γγ H, followed by the decay H bb [21, 22]. The
→ →
¯
advantage of this channel is that the amplitude for the continuum γγ bb background to
→
the Higgs signal is suppressed by a factor of mb/√sγγ when the initial-state photons
O
are in a Jz = 0 state. The production of a light(cid:0)SM Higgs(cid:1)boson through this process has
been studied in a series of papers, including the radiative QCD corrections to the signal and
to the backgrounds [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. The anticipated
experimental uncertainty in the measurement of the partial Higgs width, Γ(H γγ)
→ ×
Br(H b¯b), assuming an integrated luminosity of 80 fb 1 in the high energy peak, is about
−
→
2
2% for m < 140 GeV [6, 33, 34, 37, 38, 39, 40, 41, 42, 43].
H
¯
It is important to know that no other effect can contaminate the bb signal at the 1%
level. A possible concern studied in this paper is the interference between the resonant Higgs
¯ ¯
amplitude γγ H bb, and the continuum γγ bb process. Similar effects have been
→ → →
studied previously in gg H tt¯at a hadron collider [44], and in γγ H W+W ,
−
→ → → →
ZZ and tt¯at a photon collider [45, 46, 47]. These studies assumed a Higgs boson sufficiently
heavy that its width was at the GeV scale due to on-shell decays to W+W , ZZ and tt¯. In
−
¯
the MSSM, interference effects in γγ H bb, as well as in decays to several other final
→ →
states, were taken into account, including also Sudakov resummation [48, 49]. However,
explicit results separating out the interference contributions in the SM were not presented.
The significance ofsuch interference effects inCPasymmetries forvariouschannels ofMSSM
Higgs production and decay at a photon collider has also been explored [50]. In the case of
a light SM Higgs boson, with an MeV-scale width, the interference in gg H γγ was
→ →
considered at the LHC [51]. Resonance-continuum interference effects are usually negligible
fora narrowresonance, andform <150GeVthewidth Γ is less than17 MeVintheSM.1
H H
¯
However, the γγ H bb resonance is also rather weak, since it consists of a one-loop
→ →
production amplitude. Therefore a tree-level, or even one-loop, continuum amplitude can
potentially compete with it, especially since the tree-level mb/√sγγ suppression of the
O
¯
γγ bb continuum amplitude is absent at one loop. In theana(cid:0)logousca(cid:1)se ofgg H γγ,
→ → →
a suppression of 5% was found due to continuum interference [51].
∼
Inthe SM, theproduction amplitude γγ H proceeds at one loopand is dominated by a
→
¯
W boson in the loop, with some top quark contribution as well. The decay H bb and the
→
¯
continuum γγ bb amplitudes proceed at tree level. For m < 160 GeV, the Higgs boson
H
→
is below the tt¯and WW thresholds, so the resonant amplitude is predominantly real (i.e.,
¯
has no absorptive part), apart from the relativistic Breit-Wigner factor. The full γγ bb
→
amplitude is a sum of resonance and continuum terms,
Atotal = s−−Amγγ2H→+HAimHH→Γb¯bH +Aγγ→b¯b, (1)
where s = s is the photon-photon invariant mass. The interference term in the cross
γγ
1 In the MSSM, the widths of light Higgs bosons may be GeV-scale if tanβ is large, e.g. as considered in
ref. [50].
3
γ W,t b
b,c,τ
H + +
g
···
γ ¯b
+ +
···
FIG. 1: Feynman diagrams contributing to the interference of γγ H b¯b (upper row) with the
→ →
continuum background (lower row) up to order (α ). Only one diagram is shown at each loop
s
O
order, for each amplitude. The blob contains W and t loops, and small contributions from lighter
charged fermions.
section is given by
Re
δσγγ→H→b¯b =−2(s−m2H) (cid:8)(sA−∗γγm→H2HA)2∗H+→mb¯bA2HγΓγ2H→b¯b(cid:9) (2)
Im
+2mHΓH (sA∗γγm→H2A)2∗H+→mb¯bA2 γΓγ2→b¯b .
(cid:8) − H H H (cid:9)
Since the intrinsic Higgs width Γ is much narrower than the spread of the luminosity
H
spectrum in √s [9] and the experimental resolution δm 0.5 GeV [8], the observable
H
∼
interference effect is the integral over s across the entire linewidth. Neglecting the tiny s-
dependence of Re , the integral of the first “real” term vanishes, as
A∗γγ→HA∗H→b¯bAγγ→b¯b
it is an odd functio(cid:8)n of s around m2 . T(cid:9)he second “imaginary” term is an even function of
H
s around m2 and therefore survives the integration. However, it requires a relative phase
H
betweentheresonantandcontinuumamplitudes. Asdescribedabove,intheSMtheresonant
amplitude is mainly real, apart from the Breit-Wigner factor. The tree level continuum
¯ ¯ ¯
γγ bb amplitude is also real. The imaginary parts of the H bb and γγ bb amplitudes
→ → →
¯
arise at one loop, when we include the exchange of a gluon between the b and b quarks.
These contributions are shown schematically in fig. 1. In fact, each amplitude individually
has an infrared divergence from the soft-gluon exchange that builds up the Coulomb phase.
However, the divergence cancels in the relative phase entering Im .
A∗γγ→HA∗H→b¯bAγγ→b¯b
(cid:8) (cid:9)
4
Thus weareleft withafinitecontributiontoδσ ineq.(2). Tocompute thefractional
γγ H b¯b
→ →
interference correction to the resonance, we divide eq. (2) for δσ by the square of
γγ H b¯b
→ →
the resonant amplitude in eq. (1). We then expand all the amplitudes in α , obtaining
s
tree (1) (2) (1)
δσ
δ ≡ σγγγγ→→HH→→bb¯b¯b = 2mHΓH Im(A(γ1γA)→γHγA→btH¯bre→eb¯b "1+ AAtγγrγγe→→e bb¯¯bb − AAγ(γ1γγ)→→HH − AAtHHre→→ebb¯¯bb#), (3)
where the superscript (l) denotes the number of loops (l = 1,2) for each term in the expan-
sion, e.g. = (1) + (2) +....
Aγγ→H Aγγ→H Aγγ→H
Taking into account that thetreeamplitude tree has noabsorptive part, wecanrewrite
AH b¯b
→
δ as
2m Γ 1
δ = H H Im tree ∗tree + ∗tree (1)
tree 2 ( (1) "Aγγ→b¯bAH→b¯b AH→b¯bAγγ→b¯b
AH→b¯b Aγγ→H (4)
(2)
(cid:12)(cid:12) (cid:12)(cid:12) −Atγrγee b¯bA∗Htreeb¯bA(γ1γ)→H −Atγrγee b¯bA∗H(1)b¯b .
→ → → → #)
Aγγ H
→
(2)
We neglect the two-loop amplitude , because in the SM it is dominantly real for
Aγγ H
→
(1)
m < 2m , like , up to small contributions from loops of lighter fermions. We
H W Aγγ H
→
(1)
also separate out the contribution from the small imaginary part of , obtaining the
Aγγ H
→
expression
tree ∗tree
δ = 2mHΓH Aγγ→b¯bAH→b¯b Im (1)
tree 2"− (1) 2 Aγγ→H
AH→b¯b Aγγ→H n o (5)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) + (cid:12)(cid:12) 1 (cid:12)(cid:12) Im ∗tree (1) tree ∗(1) .
Re (1) AH→b¯bAγγ→b¯b −Aγγ→b¯bAH→b¯b #
Aγγ H n o
→
n o
The two photons in eq. (5) are taken to have identical helicity in all amplitudes, so that
J = 0 as required for interference with the production of the scalar Higgs boson.
z
We determine the imaginary parts of the two terms in the braces in the second line
of eq. (5) by analyzing the unitarity cuts of the diagram in fig. 2. The first term,
Im ∗tree (1) , comes from interpreting the b quarks crossing the left cut as the actual
AH b¯bAγγ b¯b
→ →
finanl-state b quarkos, emerging at a fixed scattering angle θ. The imaginary part of the tensor
box integral to the right of the left cut is associated with b-quark rescattering; thus one in-
tegrates over the b momenta crossing the right cut. The second term, Im tree ∗(1) ,
− Aγγ b¯bAH b¯b
→ →
comes from exchanging the roles of the left and right cuts. n o
5
γ
H
g b
γ
FIG. 2: Feynman diagram for the calculation of the interference of γγ H b¯b with the
→ →
continuum background up to order (α ). The unitarity cuts indicated by dashed vertical lines
s
O
are used to compute the imaginary parts of the various amplitudes.
We use FORM [52] for symbolic manipulations, and the decomposition of the scalar box
integralintoasix-dimensionalscalarboxplusscalartriangleintegrals[53]. Intheexpressions
below, we use the same notation as in ref. [53]; primed quantities correspond to particular
tensor integrals. After cancelling the divergent parts arising from both terms (associated
(2)
with the scalar triangle integral I [1]), the finite imaginary parts are given by
3
8Q2αα m
Im ∗tree (1)fin = b s b 2m m2 m4 6m2m2 +8m4 Im I [1]
AH b¯bAγγ b¯b m2 v b H H − b H b { 4 }
→ → H "
n o
(cid:0) (cid:1)
−8m3bm2H Im I3(4)[1] −4mb m2H −4m2b Im I3(2)′
#
n o n o
(cid:0) (cid:1)
+ cosθ cosθ , (6)
→ −
Im Atγrγee b¯bA∗H(1)bfi¯bn = 8Qm2bα2α(cid:0)vsmb −4mbm2H(cid:1)Im I3(2)′
→ → H "
n o n o
4m
+ b 2t2 +(m2 4m2)t+m2m2 +2m4 Im I(2,4)[1]
t m2 H − b b H b 2
− b n o#
(cid:2) (cid:3)
+ cosθ cosθ , (7)
→ −
(cid:0) (cid:1)
where
1
Im I [1] = c Im I(4)[1] c Im ID=6 2ǫ[1] , (8)
{ 4 } 2 4 3 − 0 4 −
πh n1+β o (cid:8) (cid:9)i
(4)
Im I [1] = ln , (9)
3 m2 1 β
n o H (cid:18) − (cid:19)
6
2(t+m2)
Im I(2)′ = π β + b , (10)
3 βm2
(cid:20) H (cid:21)
n o
(2,4)
Im I [1] = πβ, (11)
2
n o
and
(1+β)ln m2H(1+β) (1 β)ln m2H(1−β)
Im ID=6 2ǫ[1] = π 2(m2b−t) − 2(m2b−t) , (12)
4 − m2 (1+β)+h 2(t mi2) − m2 (1 β)+h 2(t mi2)
" H − b H − − b #
(cid:8) (cid:9)
2m2 (t+m2)
c = H b , (13)
4 (t m2)2(4m2 m2 )
− b b − H
t2 +t(m2 2m2)+m4
c = 4 H − b b . (14)
0 (t m2)2(4m2 m2 )
− b b − H
In the expressions above,
4m2
β 1 b , (15)
≡ s − m2
H
and
m2
t = m2 H (1+βcosθ), (16)
b − 2
¯
where θ is the γγ bb center-of-mass scattering angle. The terms in eqs. (6) and (7) that
→
are obtained by substituting cosθ cosθ (or, equivalently, t 2m2 m2 t) arise from
→ − → b− H−
a diagram like that in fig. 2, but with the two photons exchanged.
It is worth noting that the absence of bubble integrals from eq. (6) is due to a cancellation
amongthescalarandtensorbubbleterms, andthatthetensortrianglecontributionineq.(7)
(2)
has been expressed in terms of the tensor triangle integral I ′ appearing in eq. (6). After
3
(2)
adding the terms with cosθ cosθ, the contributions from I ′ drop out. Simplifying,
→ − 3
we get
Im ∗tree (1) tree ∗(1) = 32πQ2αα m2b
AH b¯bAγγ b¯b −Aγγ b¯bAH b¯b b s v
→ → → → (
n o
(m2 2m2) 1+ m2Ht (1+β)ln m2(2Hm(2b1−+tβ)) (1−β)ln m2(2Hm(2b1−−tβ))
H − b (m2 t)2 m2 (1+β)+h 2(t mi2) − m2 (1 β)+h 2(t mi2)
(cid:20) b − (cid:21) H − b H − − b
 
(m2 2m2)(t+m2) 2m2 1+β 2βm2
H − b b + b ln + b
− 2(m2 t)2 m2 1 β m2 t
(cid:20) b − H (cid:21) (cid:18) − (cid:19) b − )
+ cosθ cosθ . (17)
→ −
(cid:0) (cid:1)
7
To evaluate eq. (5), we also need the one-loop amplitude for H γγ [54, 55],
→
αm2 4m2 4m2 4m2
(1) = H 3 Q2AH q +AH τ +AH W , (18)
Aγγ H 4πv q q m2 q m2 W m2
→ " q=t,b,c (cid:18) H (cid:19) (cid:18) H (cid:19) (cid:18) H (cid:19)#
X
with
AH (x) = 2x[1+(1 x)f (x)], (19)
q −
2
AH (x) = x 3+ +3(2 x)f (x) , (20)
W − x −
(cid:20) (cid:21)
arcsin2 1 , x > 1,
√x
f (x) = (21)
2
 1 ln (cid:16)1+√(cid:17)1 x iπ , x < 1,
−
−4 1 √1 x −
− −
h (cid:16) (cid:17) i
and the tree amplitudes [21] 
m
tree = √6 b m2 4m2, (22)
AH b¯b v H − b
→
q
1 β4
tree = 8√6παQ2 − . (23)
Aγγ b¯b b1 β2cos2θ
→ −p
Here we note that the color factors have been included in eqns. (6), (7), (22) and (23); the
respective “amplitudes” are really the square roots of cross sections, summed over the b
quark colors and spins, for identical-helicity photons.
(1)
In the limit of small m , we can expand the contribution to δ coming from the
b Aγγ b¯b
→
(1)
and phases around m = 0. This approximation is excellent for almost all scattering
AH b¯b b
→
angles, because mb √sγγ. We obtain the following formula,
≪
δ 128πQ2bααsmHΓH m2 2ln 2mmHb +2ln(sinθ)+ln 11+−ccoossθθ cosθ + m4 . (24)
≈ v b (cid:0) sin(cid:1)2θ tree 2Re (cid:0)(1) (cid:1) O b
AH b¯b Aγγ H
→ → (cid:0) (cid:1)
We evaluate δ by letting α = 1/137.03(cid:12)6, α =(cid:12) 0.11(cid:8)9, v = (cid:9)246 GeV, m = 171.2 GeV,
(cid:12) s (cid:12) t
m = 4.24 GeV, m = 1.2 GeV, m = 1.78 GeV, and m = 80.4 GeV. The total Higgs
b c τ W
width Γ is computed numerically for different values of m , with results in agreement with
H H
HDECAY [56, 57].
In fig. 3 we plot δ as a function of m , for θ = 45 . We see that the interference effect is
H ◦
stronger for a heavier Higgs boson, and that it reaches 0.4% for m 150 GeV. This mass
H
− ≃
value is close to the region in which there may be sizable contributions to the phase from
W boson pairs, one on-shell and one off-shell in the H γγ amplitude; so the plot cannot
→
be extrapolated much further without performing this computation. In general, though, the
8
SM Higgs Interference Correction
0
-0.1
)
%
-0.2
(
δ
-0.3
θ = 45°
all phases turned on
γ γ → H 1-loop phase
-0.4 γ γ → b–b and H → b–b 1-loop phase
90 100 110 120 130 140 150
m (GeV)
H
FIG. 3: The percentage reduction of the SM Higgs signal as a function of the Higgs boson mass,
for center-of-mass scattering angle θ = 45 . The solid curve represents the result with all phases
◦
turned on; the dashed curves turn on different component phases each time. The effect is stronger
for a higher mass Higgs boson.
¯
dominant contribution to δ for a light Higgs boson comes from the one-loop γγ bb and
→
¯
H bb amplitudes.
→
In fig. 4 we plot δ as a function of the scattering angle θ, for m = 130 GeV. Note that
H
the small-mass approximation formula (24) for δ diverges for small angles. This behavior
¯
can be understood as coming from the γγ bb continuum amplitude, which exhibits a
→
similar angular dependence. Keeping the exact b-quark mass dependence, using eq. (17),
the divergence is regulated. We find that for m = 130 GeV, δ = 18% at θ = 3 , and
H ◦
that it rolls off to a constant δ 35% for θ < 0.5 . Of course it would be very challenging
◦
≈
experimentally to search for b jets in this far-forward region, and the reason δ is increasing
¯
is because the continuum bb background is increasing. Away from the forward region, the
interference effect has the opposite sign, negative, and its magnitude becomes maximum for
9
SM Higgs Interference Correction
0.05
0
-0.05
)
%
(
δ
-0.1
-0.15
m = 130 GeV
H
all phases turned on
γ γ → H 1-loop phase
γ γ → b–b and H → b–b 1-loop phase
-0.2
20 30 40 50 60 70 80 90
θ (degrees)
FIG. 4: The percentage reduction of the SM Higgs signal as a function of the scattering angle for
m = 130 GeV. The solid curve represents the result with all phases turned on; the dashed curves
H
turn on different component phases each time. The total effect is maximized close to θ 35 .
◦
≃
¯ ¯
θ 35 , with δ 0.18%. Again, the phase arising from the one-loop γγ bb and H bb
◦
≃ ≃ − → →
amplitudes almost solely determines the size of the correction.
In models beyond the SM, such as the MSSM, the coupling of a Higgs boson to b quarks
and to photons is modified. How does the interference effect depend on these couplings?
Looking at eq. (24), we see that the two powers of the Yukawa coupling λ m /v from
b b
≡
tree 2 cancel against the ones contained in Γ (which for most of the relevant range of
AH b¯b H
→
¯
(cid:12)m is d(cid:12)ominated by the H bb decay). There is one extra power of λ = m /v coming
H b b
(cid:12) (cid:12) →
¯
from the H bb amplitude in the numerator in eq. (5), so the dominant contribution to δ
→
(1)
is linear in λ . The subdominant contribution from Im includes one more factor of
b {Aγγ H}
→
λ , so it is quadratic in λ .
b b
At a photon collider, the unperturbed peak height is proportional to the product Γ(H
→
¯
γγ) Br(H bb). The H γγ width does not depend strongly on λ until it gets very
b
× → →
10