Linearly Polarized Gluons and the Higgs Transverse Momentum Distribution

0.33 MB·English

by  Daniel Boer

Linearly Polarized Gluons and the Higgs Transverse Momentum Distribution
Dani¨el Boer,1 Wilco J. den Dunnen,2 Cristian Pisano,3 Marc Schlegel,4 and Werner Vogelsang4
1Theory Group, KVI, University of Groningen, Zernikelaan 25, NL-9747 AA Groningen, The Netherlands
2Department of Physics and Astronomy, Vrije Universiteit Amsterdam, NL-1081 HV Amsterdam, The Netherlands
3Dipartimento di Fisica, Universita` di Cagliari, and INFN,
Sezione di Cagliari, I-09042 Monserrato (CA), Italy
4Institute for Theoretical Physics, Universita¨t Tu¨bingen,
Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
(Dated: January 20, 2012)
Westudyhowgluonscarryinglinearpolarizationinsideanunpolarizedhadroncontributetothe
transversemomentumdistributionofHiggsbosonsproducedinhadroniccollisions. Theymodifythe
distributionproducedbyunpolarizedgluonsinacharacteristicwaythatcouldbeusedtodetermine
whether the Higgs boson is a scalar or a pseudoscalar particle.
2
1 PACSnumbers: 12.38.-t;13.85.Ni;13.88.+e
0
2
It is sometimes said that the LHC is a ‘gluon collider’, double helicity flip involved (see Fig. 1). While this may

n
becauseathighenergiesthegluondensityinsideaproton be expected to make only a relatively modest contribu-
a
becomes dominant over the quark densities. Higgs pro- tion, the function h⊥g is of non-perturbative nature and
J 1
duction, in particular, predominantly arises from gluon- is present at tree level already. Therefore, a significant
9
1 gluon ‘fusion’ gg H through a triangular top quark influence of linearly polarized gluons on the distribution
→
loop. QCD corrections to this process have been cal- of the produced particle at low q is not excluded.
T
] culated with increasing precision [1–7], making it well
h
p understood. Itisnotcommonlyknownhoweverthatthe
- LHCisactuallyalsotosomeextentapolarizedgluoncol- f1g h⊥1 g
p
lider, since gluons can in principle be linearly polarized
e
h insideanunpolarizedproton. Theircorrespondingdistri- 1 1 1 1
± ± ± ∓
[ bution, heredenotedbyh⊥g andfirstdefinedinRef.[8],
1
requiresthegluonstohaveanonzerotransversemomen-
2 1 1 1 1
tumwithrespecttotheparenthadron. Itcorrespondsto ± ± ± ∓
v
4 an interference between +1 and −1 helicity gluon states f1g h⊥1 g
4 thatwouldbesuppressedwithouttransversemomentum.
4
1 So far the function h⊥g has not been studied exper-
1
. imentally, and consequently nothing is known about its FIG.1: Gluonhelicitiesinthegg→H squaredamplitudefor
9
unpolarized (left) and linearly polarized production (right).
0 magnitude. Only a theoretical upper bound has been
1 given [8, 9]. Recently, several ways of probing h⊥g have
1
1 been put forward, namely in heavy quark pair or dijet In light of this, we will investigate in this letter how
v: production [9], or in photon pair production [10], where the distribution of linearly polarized gluons may affect
i in all cases the transverse momentum of the pair is mea- the transverse momentum distribution of Higgs bosons
X
sured. One way in which linearly polarized gluons can forq m ,wherem istheHiggsmass. Weshallob-
r T (cid:28) H H
a manifest themselves in these processes is through az- serve that linearly polarized gluons may in fact provide
imuthal asymmetries. However, it was found that they a tool to uncover whether the Higgs boson is a scalar or
canalsogenerateaterminthecrosssectionthatisinde- a pseudoscalar particle. Thus far, relatively few sugges-
pendent of azimuthal angle. This happens when two lin- tionstothisendhavebeenputforwardfortheLHC,typ-
earlypolarizedgluons,onefromeachhadron,participate icallyusingazimuthaldistributions,forexampleinHiggs
inthescattering. Inthiswaytheycanalsocontributeto + jet pair production [13] or in τ pair decays [14]. The
production of a scalar particle, such as a scalar or pseu- suggestionweputforwardheredoesnotinvolvemeasure-
doscalar Higgs boson, when its transverse momentum q mentsofanyangulardistributions. Instead,wewillshow
T
ismeasured. Ithasinfactbeenshown[11,12]thatsucha that linear polarization of gluons simply leads to a mod-
contributionisgeneratedperturbatively. Inotherwords, ulation of the Higgs transverse momentum distribution
ifattreelevelgluonsaretakentobeunpolarized,atorder that depends on the nature of the Higgs particle.
α theywillbecometosomeextentlinearlypolarized. In Transverse momentum dependent distribution func-
s
thetransversemomentumdistributionofspin-0particles tions (TMDs) of gluons in an unpolarized hadron are
produced in proton-proton collisions this will give rise to defined through a matrix element of a correlator of the
an additional contribution at order α2, because of the gluon field strengths Fµν(0) and Fνσ(ξ), evaluated at
s
2
fixed light-front (LF) time ξ+ = ξ n = 0, where n is ForapseudoscalarHiggsbosonA0withasimplecoupling
·
a lightlike vector conjugate to the parent hadron’s four- g (√2G )1/2m2γ to quarks [19] we have instead
t F t 5
momentum P. Decomposing the gluon momentum as
p=xP +pT +p−n, the correlator is given by [8] MˆµAν =i21/4G1F/2αsm2H(cid:15)µTνAA(τ)/(8π), (5)
Φµgν(x,pT) = (npρnn)σ2(cid:90) d(ξ(·P2π))d32ξT eip·ξ whAe(rτe)(cid:15)=µTνgits2tJh(eτ)tw/τo.-dAimsemnseinotnioanleLdevaib-Covivei,taQtCeDnsocro,rraencd-
A
· tionstotheseamplitudeshavebeencalculated. However,
P Tr(cid:2)Fµρ(0)Fνσ(ξ)(cid:3) P (cid:5)
×(cid:104) | | (cid:105) LF since our goal is to study the effect of linearly polarized
1 (cid:26) (cid:18)pµpν p2 (cid:19) (cid:27) gluons whose distribution h⊥g is anyway unknown, we
= gµνfg T T +gµν T h⊥g ,(1) 1
−2x T 1 − M2 T 2M2 1 limit ourselves to the lowest-order expressions (3) and
(5). This leads to the following expressions for scalar
withp2 = p2,gµν =gµν Pµnν/P n nµPν/P n,and and pseudoscalar Higgs production:
M theTpro−tonTmaTss. fg(x−,p2) repr·ese−nts the un·polar-
iozfedlingelaurolny pdiosltarriibzuetdiognlu1aonnds.hT⊥1Ing((x1,)pw2T)ethhaevediostmriibttuetdiona Eddσ3H(cid:126)q(A)(cid:12)(cid:12)(cid:12)qT(cid:28)mH = 1π2√8m2G2HFS (cid:16)4απs(cid:17)2 (cid:12)(cid:12)AH(A)(τ)(cid:12)(cid:12)2
(cid:16) (cid:104) (cid:105)(cid:17) q
Wilson line that renders the correlator gauge invariant. [fgfg] w h⊥gh⊥g + ( T ),(6)
As any TMD, h⊥g will receive contributions from initial × C 1 1 ±C H 1 1 O mH
1
and/or final state interactions, which make the gauge where the upper (lower) sign refers to the scalar (pseu-
link process-dependent. Therefore, despite the fact that doscalar) case, and where we have the TMD convolution
itisT-even,h⊥g canreceivenon-universalcontributions,
1 (cid:90) (cid:90)
and its extraction can be hampered for processes where [wff] d2p d2p δ2(p +p q )
factorization does not hold, such as dijet production in C ≡ aT bT aT bT − T
hadron-hadron collisions [15–17]. Higgs production, on w(p ,p )f(x ,p2 )f(x ,p2 ), (7)
× aT bT a aT b bT
the other hand, is expected to allow for TMD factoriza-
with the transverse momentum weight
tion, just like the Drell-Yan process. A more detailed
study of this remains to be carried out. (p p )2 1p2 p2
The calculation of the Higgs production cross section wH = aT · bT2M−4 2 aT bT . (8)
intheTMDframeworkcloselyfollowsRefs.[15,18]. The
generic contribution by gg H reads We emphasize the sign difference in the [w h⊥gh⊥g]
→ C H 1 1
term in Eq. (6), which may offer an opportunity to de-
E dσ(cid:12) πx x
H (cid:12) = a b (2) terminetheparityoftheHiggsboson. Thetermsinvolv-
(cid:90)d3(cid:126)q (cid:12)qT(cid:90)(cid:28)mH 16m2HS × i(cid:82)ndg2hq⊥1g(qh2a)vαe th[we mho⊥dgelh-i⊥ngd]ep=en0dfeonrtαpro=pe0r,t1y.(cid:104)qT2Thαi(cid:105)shfhea≡-
d2paT d2pbT δ2(paT +pbT −qT)Φµgν(xa, paT) ture pToinTts toCwarHds1a di1stinctive transverse momentum
Φρσ(x , p ) (cid:16) ˆµρ(cid:17)(cid:16) ˆνσ(cid:17)∗(cid:12)(cid:12)pb=xbPb + (cid:18) qT (cid:19). distribution of the h⊥1gh⊥1g term with a double node in
× g b bT M M (cid:12)pa=xaPa O mH qT. We note that since (cid:104)1(cid:105)hh = 0, linearly polarized
gluons do not affect the q -integrated cross section.
T
For now, we assume on-shell production of the Higgs In the following we estimate the possible size of the
particle, with (cid:126)q and E its momentum and energy. contribution by linearly polarized gluons to the Higgs
H
P and P are the momenta of the colliding protons, productioncrosssectionattreelevel. Althoughthefunc-
a b
S = (P + P )2, and x = q2/(2P q). To low- tion h⊥g itself is unknown, a model-independent positiv-
a b a(b) a(b) 1
·
estorder,thehardpartonicamplitude ˆ isgivenbythe ity bound for it has been derived in Ref. [8]:
M
well-knownformula[4]forthegg H trianglediagram:
→ p2T h⊥g(x,p2) fg(x,p2), (9)
ˆµν =i21/4G1/2α m2 gµν (τ)/(8π) (3) 2M2 | 1 T |≤ 1 T
MH F s H T AH validforallxandp . Themaximallypossibleeffectwill
T
for a scalar Standard Model (SM) Higgs boson H0, be generated when this bound is saturated. Models may
where we only consider top quarks in the triangle, and also shed light on the size of h⊥g. In the simple pertur-
1
where G is the Fermi constant, α the strong coupling bativequarktargetmodelofgluonTMDsofRef.[20]the
F s
constant, τ = m2 /(4m2) with the top mass m , and function h⊥g is found to possess the same characteristic
H t t 1
(τ)=2(τ +(τ 1)J(τ))/τ2 with 1/xincreaseasthedistributionofunpolarizedgluonsfg,
AH − 1
which suggests that linearly polarized gluons may be as
 1(cid:18)ln(cid:18)1+√1−1/τ(cid:19) iπ(cid:19)2 , τ >1 relevant at small x as unpolarized ones. Another recent
J(τ)= −4 1−√1−1/τ − (4) model calculation [21] shows saturation of the positivi-
 arcsin2(√τ) , τ 1. ty bound for h⊥g in heavy nuclei in certain transverse
≤ 1
3
momentumregions(Weizs¨acker-Williamsmodel)oreven the effect enters scalar and pseudoscalar Higgs produc-
over the full momentum range (dipole model). This sug- tionwithoppositesign. Iftheeffectisatornearthelevel
gests that saturation of the positivity bound at least lo- shown by Fig. 2, it should easily allow to determine the
cally in x or p2 might not be an unrealistic assumption. parityoftheHiggsboson,providedasufficientlyfinescan
T
WefollowastandardapproachforTMDsinthelitera- in q is possible in experiment, to resolve both nodes.
T
ture(see[22])andassumeasimpleGaussiandependence So far we have considered only the production of an
of the gluon TMDs on transverse momentum: on-shellHiggsboson. InrealitytheHiggswilldecayinto
some observed final state, and there will be background
fg(x,p2)= G(x) exp(cid:18) p2T (cid:19) , (10) reactions contributing to this final state that are not re-
1 T π p2 − p2 lated to the Higgs. These backgrounds may themselves
(cid:104) T(cid:105) (cid:104) T(cid:105)
be sensitive to linearly polarized gluons. We will now
where G(x) is the collinear gluon distribution and the brieflyconsideroneexampleofthis,theHiggsdecayinto
width p2 isassumedtobeindependentofx. Thebound a photon pair. We reserve a more detailed study of final
(cid:104) T(cid:105)
(9) is directly satisfied by the form states such as γZ, ZZ or WW for a future publication.
After production in gg H, the two-photon decay
M2G(x)2e(1 r) (cid:18) p2 (cid:19) →
h⊥g(x,p2)= − exp T . (11) of a SM Higgs occurs through a top quark or W-boson
1 T π p2 2 r −r p2 triangular loop. The decay of a pseudoscalar Higgs is
(cid:104) T(cid:105) (cid:104) T(cid:105)
insteaddescribedbyphysicsbeyondtheSMandhenceis
We choose r = 2/3. The left panel of Fig. 2 shows the
model-dependent. Thereareoftennotree-levelcouplings
p -dependenceoffg andh⊥g fortwovaluesoftheGaus-
T 1 1 to W-bosons in this case [24], so here we consider only
sian width: p2 = 1GeV2 and p2 = 7GeV2. The
(cid:104) T(cid:105) (cid:104) T(cid:105) thetopquarkcoupling. Forbothascalarorpseudoscalar
lattervaluemaybemoreappropriateatQ=m ,cf.the
H Higgs, the lowest-order amplitude can be written as [25]
Gaussianfittofu(x,p2)evolvedtoQ=M ofRef.[23].
1 T Z
√2G α αs2 (τ¯) (s)
ˆµν = F s AH(A) FH(A)→γγ rµν, (14)
0000....33335555 000...666 Mγγ 32π2 s−m2H +iΓHmH
RRR
0000 .0.0.0.02222....53535353 ppppT2T2T2T2 //// ((((2222MMMM2222)))) **** hhhh1⊥1⊥1⊥1⊥ffff1111 gggggggg 000000......454545 〈〈〈〈〈〈 ppppppT2T2T2T2T2T2 〉〉〉〉〉〉== 71 GGeeVV22 wdehcearyewαiidstthh.eFeolercatrsocmalaagrnHeitgicgsc,oruµpνli=nggaTµnνdδλΓaHλbt,hwehHerigegass
000...333 rµν = λ i(cid:15)µνδ in the pseudoscalar case, with λ , λ
0000....2222 a T λaλb a b
〈〈〈〈 ppppT2T2T2T2 〉〉〉〉 ====1111 GGGGeeeeVVVV2222 000...222 the photon helicities. H(τ¯) and A(τ¯) are given in
0000....11115555 000...111 Eqs. (3) and (5) with τ¯A= s/(4m2t)A, where s = (pa +
0000....1111 1111 p )2 x x S for gluon momenta p , p . Finally,
000 b a b a b
(cid:39)
0000....00005555 7777 ---000...111 4
7777 (s)= (τ )+ N (τ¯), (15)
0000 0000 0000....5555 1111 1111....5555 2222 2222....5555 3333 3333....5555 4444 ---000...222 000 222 444 666 888 111000 111222 111444 111666 FH→γγ W W 9 cAH
ppppTTTT (((( GGGG eeee VVVV )))) qqqTTT ((( GGG eee VVV ))) with τ = s/(4m2 ) and (τ) = (2τ2+3τ +3(2τ
W W W − −
1)J(τ))/τ2 describes the contribution by the W trian-
FIG.2: Left: Gaussian distributions for fg and h⊥g (divided
by G(x)) as functions of pT for two differ1ent valu1es of (cid:104)p2T(cid:105). gular loop. We assume FA→γγ(s) = 94NcAA(τ¯). In the
Right: Resulting ratio R=C[wHh⊥1gh⊥1g]/C[f1gf1g]. 1fo2l0loGweinVgwwiethcoanssmidaelrltaotraellawtiivdetlhyΓlightH5ig1g0s−m3aGsseVm[H26=].
H
(cid:39) ×
As is well known, an important QCD background to
It is straightforward to compute the convolution inte-
photon pair production at high energies is generated by
grals appearing in Eq. (6) analytically:
gg γγ via a quark box [27]. This subprocess was
→
G(x )G(x ) (cid:18) q2 (cid:19) studied recently in the context of TMD factorization in
C[f1gf1g]= 2πa p2 b exp −2 pT2 , (12) Ref. [10]. Using its results, we add the two lowest-order
(cid:104) T(cid:105) (cid:104) T(cid:105) amplitudes describing the box diagram and the Higgs
G(x )G(x )
[w h⊥gh⊥g]= a b resonance and extract the azimuthal-angle independent
C H 1 1 36π p2 × cross section:
(cid:104) T(cid:105)
(cid:20)23 − (cid:104)qp2T2T(cid:105) + 136((cid:104)qp2T2T)(cid:105)22(cid:21) exp(cid:18)2− 43(cid:104)qp2T2T(cid:105)(cid:19) . (13) (cid:90) dφdd4σqgdgΩ =F1C[f1gf1g]+F2C[wHh⊥1gh⊥1g]. (16)
TheirratioRisshownintherightpanelofFig.2. Itis Here q = q +q is the momentum of the photon pair.
a b
a measure of the relative size of the contribution by lin- dΩ = dφdcosθ denotes the solid angle element for each
early polarized gluons and shows the anticipated double photon, with the angles φ,θ defined in the Collins-Soper
node of [w h⊥gh⊥g]. It is evident that at least within frame [10]. F and F are calculated functions of θ and
C H 1 1 1 2
oursimplemodel,linearlypolarizedgluonshaveasizable the pair mass Q = √s that we will not give here. They
effect on the Higgs q distribution. We stress again that depend on the box and Higgs amplitudes.
T
4
scalar pseudoscalar SardegnaundergrantPOSardegnaFSE2007-2013,L.R.
1
7/2007. This research is part of the FP7 EU-programme
q = p/2
0.8q = 3p/8 0.2 Hadron Physics (No. 227431) and part of the research
0.6 q = p/4 119.7 mH 120.3 0 programofthe“StichtingvoorFundamenteelOnderzoek
q = p/8 der Materie (FOM)” which is financially supported by
0.4
F1 -0.2 the“NederlandseOrganisatievoorWetenschappelijkOn-
/2 0.2 derzoek (NWO)”. W.V.’s work is supported by the U.S.
F 119.7 mH 120.3 -0.4
0 Department of Energy (contract DE-AC02-98CH10886).
-0.6
-0.2
-0.8
-0.4
-1
Q [GeV] Q [GeV]
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We thank John Collins and Feng Yuan for useful dis-
(2011).
cussions. C.P. is supported by Regione Autonoma della