Resonance-continuum interference in light Higgs boson production at a photon collider

Lance J. Dixon and Yorgos Sofianatos

SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94309, USA(Received 7 January 2009; published 3 February 2009)

We study the effect of interference between the standard model Higgs boson resonance and the

continuum background in the process �� ! H ! b �b at a photon collider. Taking into account virtual

gluon exchange between the final-state quarks, we calculate the leading corrections to the height of the

resonance for the case of a light (mH < 160 GeV) Higgs boson. We find that the interference is destructive

and around 0.1%–0.2% of the peak height, depending on the mass of the Higgs and the scattering angle.

This suppression is smaller by an order of magnitude than the anticipated experimental accuracy at a

photon collider. However, the fractional suppression can be significantly larger if the Higgs coupling to b

quarks is increased by physics beyond the standard model.

DOI: 10.1103/PhysRevD.79.033002 PACS numbers: 14.80.Bn, 13.66.Fg, 14.80.Cp

The standard model of particle physics (SM) has beenvery successful in describing a wide range of elementaryparticle phenomena to high accuracy. A key ingredient ofthe model is the scalar Higgs field, responsible for electro-weak symmetry breaking and for generating the masses ofessentially all massive elementary particles [1–3]. Similarfields exist in extensions of the SM, such as the minimalsupersymmetric standard model (MSSM). In the SM, theHiggs boson is the only particle that remains undiscovered,and its properties are determined by its mass. It is a maingoal of current and future high energy physics experimentsto identify the Higgs boson and explore the details of theHiggs sector. In particular, the discovery of the Higgsboson could take place at run II of the Tevatron atFermilab; if not there, then at the Large Hadron Collider(LHC) at CERN. Precise measurements of its propertieswill be one of the tasks of the proposed InternationalLinear Collider (ILC). There is an option to use the ILCas a photon collider, by backscattering laser light off of thehigh energy electron beams. The high energy, highly po-larized photons produced in this way can be used to studythe various Higgs couplings to very high accuracy [4–9].

The mass of the Higgs boson in the SM and MSSM hasalready been constrained by experiment to a range wellwithin the reach of the aforementioned designed machines.Precision electroweak measurements have put an upperbound on the allowed values for its mass, mH &170 GeV at 95% confidence level in the SM [10,11]. Inthe MSSM the Higgs boson mass obeys the bound mH �mZ at tree level; radiative corrections increase this limit toabout 135 GeV [12–14]. The mass of the Higgs boson hasalso been bounded from below via the Higgs-strahlungprocess eþe� ! HZ at LEP2, with mH * 114:1 GeV inthe SM and mH * 91:0 GeV in the MSSM [15–20].

At a photon collider, among the two possible modes, ��and e�, the former is especially useful for Higgs physics.For mH < 140 GeV, the most important channel involvesHiggs production via photon fusion, �� ! H, followed bythe decay H ! b �b [21,22]. The advantage of this channel

is that the amplitude for the continuum �� ! b �b back-ground to the Higgs signal is suppressed by a factor ofOðmb=

ffiffiffiffiffiffiffis��

p Þ when the initial-state photons are in a Jz ¼ 0

state. The production of a light SM Higgs boson throughthis process has been studied in a series of papers, includ-ing the radiative QCD corrections to the signal and to thebackgrounds [23–36]. The anticipated experimental uncer-tainty in the measurement of the partial Higgs width,�ðH ! ��Þ � BrðH ! b �bÞ, assuming an integrated lumi-nosity of 80 fb�1 in the high energy peak, is about 2% formH < 140 GeV [6,33,34,37–43].It is important to know that no other effect can contami-

nate the b �b signal at the 1% level. A possible concernstudied in this paper is the interference between the reso-nant Higgs amplitude �� ! H ! b �b, and the continuum�� ! b �b process. Similar effects have been studied pre-viously in gg ! H ! t�t at a hadron collider [44], and in�� ! H ! WþW�, ZZ, and t�t at a photon collider [45–47]. These studies assumed a Higgs boson sufficientlyheavy that its width was at the GeV scale due to on-shelldecays to WþW�, ZZ, and t�t. In the MSSM, interferenceeffects in �� ! H ! b �b, as well as in decays to severalother final states, were taken into account, including alsoSudakov resummation [48,49]. However, explicit resultsseparating out the interference contributions in the SMwere not presented. The significance of such interferenceeffects in CP asymmetries for various channels of MSSMHiggs production and decay at a photon collider has alsobeen explored [50]. In the case of a light SM Higgs boson,with an MeV-scale width, the interference in gg ! H !��was considered at the LHC [51]. Resonance-continuuminterference effects are usually negligible for a narrowresonance, and for mH < 150 GeV the width �H is lessthan 17 MeV in the SM.1 However, the �� ! H ! b �bresonance is also rather weak, since it consists of a one-loop production amplitude. Therefore a tree-level, or even

1In the MSSM, the widths of light Higgs bosons may be GeVscale if tan� is large, e.g. as considered in Ref. [50].

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one-loop, continuum amplitude can potentially competewith it, especially since the tree-level Oðmb=

ffiffiffiffiffiffiffis��

p Þ sup-pression of the �� ! b �b continuum amplitude is absent atone loop. In the analogous case of gg ! H ! ��, asuppression of �5% was found due to continuum interfer-ence [51].

In the SM, the production amplitude �� ! H proceedsat one loop and is dominated by aW boson in the loop, withsome top quark contribution as well. The decay H ! b �band the continuum �� ! b �b amplitudes proceed at treelevel. For mH < 160 GeV, the Higgs boson is below the t�tandWW thresholds, so the resonant amplitude is predomi-nantly real (i.e., has no absorptive part), apart from therelativistic Breit-Wigner factor. The full �� ! b �b ampli-tude is a sum of resonance and continuum terms,

A total ¼�A��!HAH!b �b

s�m2H þ imH�H

þA��!b �b; (1)

where s ¼ s�� is the photon-photon invariant mass. The

interference term in the cross section is given by

����!H!b �b ¼ �2ðs�m2HÞ

� RefA���!HA

�H!b �b

A��!b �bgðs�m2

HÞ2 þm2H�

2H

þ 2mH�H

ImfA���!HA

�H!b �b

A��!b �bgðs�m2

HÞ2 þm2H�

2H

:

(2)

Since the intrinsic Higgs width �H is much narrowerthan the spread of the luminosity spectrum in

ffiffiffis

p[9] and

the experimental resolution �mH � 0:5 GeV [8], the ob-servable interference effect is the integral over s across theentire linewidth. Neglecting the tiny s dependence ofRefA�

��!HA�H!b �b

A��!b �bg, the integral of the first

‘‘real’’ term vanishes, as it is an odd function of s aroundm2

H. The second ‘‘imaginary’’ term is an even function of saround m2

H and therefore survives the integration.However, it requires a relative phase between the resonantand continuum amplitudes. As described above, in the SMthe resonant amplitude is mainly real, apart from the Breit-Wigner factor. The tree-level continuum �� ! b �b ampli-tude is also real. The imaginary parts of the H ! b �b and�� ! b �b amplitudes arise at one loop, when we includethe exchange of a gluon between the b and �b quarks. Thesecontributions are shown schematically in Fig. 1. In fact,each amplitude individually has an infrared divergencefrom the soft-gluon exchange that builds up the Coulombphase. However, the divergence cancels in the relativephase entering ImfA�

��!HA�H!b �b

A��!b �bg. Thus we

are left with a finite contribution to ����!H!b �b in Eq.

(2). To compute the fractional interference correction to theresonance, we divide Eq. (2) for ����!H!b �b by the square

of the resonant amplitude in Eq. (1). We then expand all theamplitudes in �s, obtaining

� � ����!H!b �b

���!H!b �b

¼ 2mH�H Im

� Atree��!b �b

Að1Þ��!HA

treeH!b �b

�1þ

Að1Þ��!b �b

Atree��!b �b

�Að2Þ��!H

Að1Þ��!H

�Að1ÞH!b �b

AtreeH!b �b

��; (3)

where the superscript (l) denotes the number of loops (l ¼1, 2) for each term in the expansion, e.g. A��!H ¼Að1Þ

��!H þAð2Þ��!H þ � � � .

Taking into account that the tree amplitude AtreeH!b �b

has

no absorptive part, we can rewrite � as

� ¼ 2mH�H

jAtreeH!b �b

j2 Im

�1

Að1Þ��!H

�Atree

��!b �bA�tree

H!b �b

þA�treeH!b �b

Að1Þ��!b �b

�Atree��!b �b

A�treeH!b �b

Að2Þ��!H

Að1Þ��!H

�Atree��!b �b

A�ð1ÞH!b �b

��: (4)

We neglect the two-loop amplitudeAð2Þ��!H, because in the

SM it is dominantly real for mH < 2mW , like Að1Þ��!H, up

to small contributions from loops of lighter fermions. Wealso separate out the contribution from the small imaginary

part of Að1Þ��!H, obtaining the expression

� ¼ 2mH�H

jAtreeH!b �b

j2��

Atree��!b �b

A�treeH!b �b

jAð1Þ��!Hj2

ImfAð1Þ��!Hg

þ 1

RefAð1Þ��!Hg

ImfA�treeH!b �b

Að1Þ��!b �b

�Atree��!b �b

A�ð1ÞH!b �b

g�: (5)

The two photons in Eq. (5) are taken to have identical

FIG. 1 (color online). Feynman diagrams contributing to theinterference of �� ! H ! b �b (upper row) with the continuumbackground (lower row) up to order Oð�sÞ. Only one diagram isshown at each loop order, for each amplitude. The blob containsW and t loops, and small contributions from lighter chargedfermions.

LANCE J. DIXON AND YORGOS SOFIANATOS PHYSICAL REVIEW D 79, 033002 (2009)

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helicity in all amplitudes, so that Jz ¼ 0 as required forinterference with the production of the scalar Higgs boson.

We determine the imaginary parts of the two terms in thebraces in the second line of Eq. (5) by analyzing theunitarity cuts of the diagram in Fig. 2. The first term,

ImfA�treeH!b �b

Að1Þ��!b �b

g, comes from interpreting the b

quarks crossing the left cut as the actual final-state bquarks, emerging at a fixed scattering angle �. The imagi-nary part of the tensor box integral to the right of the leftcut is associated with b-quark rescattering; thus one inte-grates over the b momenta crossing the right cut. The

second term, �ImfAtree��!b �b

A�ð1ÞH!b �b

g, comes from ex-

changing the roles of the left and right cuts.We use FORM [52] for symbolic manipulations, and the

decomposition of the scalar box integral into a six-dimensional scalar box plus scalar triangle integrals [53].In the expressions below, we use the same notation as inRef. [53]; primed quantities correspond to particular tensorintegrals. After canceling the divergent parts arising fromboth terms (associated with the scalar triangle integral

Ið2Þ3 ½1�), the finite imaginary parts are given by

ImfA�treeH!b �b

Að1Þfin��!b �b

g

¼ 8Q2b��smb

m2Hv

½2mbm2Hðm4

H � 6m2bm

2H

þ 8m4bÞ ImfI4½1�g � 8m3

bm2H ImfIð4Þ3 ½1�g

� 4mbðm2H � 4m2

bÞ ImfIð2Þ03 g� þ ðcos� ! � cos�Þ;(6)

ImfAtree��!b �b

A�ð1ÞfinH!b �b

g ¼ 8Q2b��smb

m2Hv

��4mbm

2H ImfIð2Þ03 g

þ 4mb

t�m2b

½2t2 þ ðm2H � 4m2

bÞt

þm2bm

2H þ 2m4

b� ImfIð2;4Þ2 ½1�g�

þ ðcos� ! � cos�Þ; (7)

where

Im fI4½1�g ¼ 12½c4 ImfIð4Þ3 ½1�g � c0 ImfID¼6�2�

4 ½1�g�; (8)

Im fIð4Þ3 ½1�g ¼ �

m2H

ln

�1þ �

1� �

�; (9)

Im fIð2Þ03 g ¼ �

��þ 2ðtþm2

b�m2

H

�; (10)

Im fIð2;4Þ2 ½1�g ¼ ��; (11)

and

Im fID¼6�2�4 ½1�g ¼ �

� ð1þ �Þ ln½m2Hð1þ�Þ2ðm2

b�tÞ �

m2Hð1þ �Þ þ 2ðt�m2

bÞ

�ð1� �Þ ln½m2

Hð1��Þ2ðm2

b�tÞ �

m2Hð1� �Þ þ 2ðt�m2

b�; (12)

c4 ¼ 2m2Hðtþm2

bÞðt�m2

bÞ2ð4m2b �m2

HÞ; (13)

c0 ¼ 4t2 þ tðm2

H � 2m2bÞ þm4

b

ðt�m2bÞ2ð4m2

b �m2HÞ

: (14)

In the expressions above,

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4m2

b

m2H

s; (15)

and

t ¼ m2b �

m2H

2ð1þ � cos�Þ; (16)

where � is the �� ! b �b center-of-mass scattering angle.The terms in Eqs. (6) and (7) that are obtained by sub-stituting cos� ! � cos� (or, equivalently, t ! 2m2

b �m2

H � t) arise from a diagram like that in Fig. 2, but withthe two photons exchanged.It is worth noting that the absence of bubble integrals

from Eq. (6) is due to a cancellation among the scalar andtensor bubble terms, and that the tensor triangle contribu-tion in Eq. (7) has been expressed in terms of the tensor

triangle integral Ið2Þ03 appearing in Eq. (6). After adding the

terms with cos� ! � cos�, the contributions from Ið2Þ03

drop out. Simplifying, we get

FIG. 2 (color online). Feynman diagram for the calculation ofthe interference of �� ! H ! b �b with the continuum back-ground up to orderOð�sÞ. The unitarity cuts indicated by dashedvertical lines are used to compute the imaginary parts of thevarious amplitudes.

RESONANCE-CONTINUUM INTERFERENCE IN LIGHT . . . PHYSICAL REVIEW D 79, 033002 (2009)

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ImfA�treeH!b �b

Að1Þ��!b �b

�Atree��!b �b

A�ð1ÞH!b �b

g ¼ 32�Q2b��s

m2b

v

�ðm2

H � 2m2bÞ�1þ m2

Ht

ðm2b � tÞ2

�� ð1þ �Þ ln½m2Hð1þ�Þ2ðm2

b�tÞ �

m2Hð1þ �Þ þ 2ðt�m2

bÞ

�ð1� �Þ ln½m2

Hð1��Þ2ðm2

b�tÞ �

m2Hð1� �Þ þ 2ðt�m2

b��

�ðm2H � 2m2

bÞðtþm2bÞ

2ðm2b � tÞ2 þ 2m2

b

m2H

�ln

�1þ �

1� �

�

þ 2�m2b

m2b � t

�þ ðcos� ! � cos�Þ: (17)

To evaluate Eq. (5), we also need the one-loop amplitudefor H ! �� [54,55],

Að1Þ��!H ¼ �m2

H

4�v

�3

Xq¼t;b;c

Q2qA

Hq

�4m2

q

m2H

�þ AH

q

�4m2

m2H

�

þ AHW

�4m2

W

m2H

��; (18)

with

AHq ðxÞ ¼ 2x½1þ ð1� xÞfðxÞ�; (19)

AHWðxÞ ¼ �x

�3þ 2

xþ 3ð2� xÞfðxÞ

�; (20)

fðxÞ ¼� arcsin2ð 1ffiffi

xp Þ; x 1;

� 14 ½lnð1þ

ffiffiffiffiffiffiffi1�x

p1� ffiffiffiffiffiffiffi

1�xp Þ � i��2; x < 1;

(21)

and the tree amplitudes [21]

A treeH!b �b

¼ ffiffiffi6

p mb

v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

H � 4m2b

q; (22)

A tree��!b �b

¼ 8ffiffiffi6

p��Q2

b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �4

p1� �2cos2�

: (23)

Here we note that the color factors have been included inEqs. (6), (7), (22), and (23); the respective ‘‘amplitudes’’are really the square roots of cross sections, summed overthe b quark colors and spins, for identical-helicity photons.

In the limit of small mb, we can expand the contribution

to � coming from theAð1Þ��!b �b

andAð1ÞH!b �b

phases around

mb ¼ 0. This approximation is excellent for almost allscattering angles, because mb ffiffiffiffiffiffiffi

s��p

. We obtain the

following formula:

� � 128�Q2b��smH�H

vm2

b

� 2 lnðmH

2mbÞ þ 2 lnðsin�Þ þ lnð1�cos�

1þcos�Þ cos�sin2�jAtree

H!b �bj2RefAð1Þ

��!HgþOðm4

bÞ:

(24)

We evaluate � by letting � ¼ 1=137:036, �s ¼ 0:119,v ¼ 246 GeV, mt ¼ 171:2 GeV, mb ¼ 4:24 GeV, mc ¼

1:2 GeV,m ¼ 1:78 GeV, andmW ¼ 80:4 GeV. The totalHiggs width �H is computed numerically for differentvalues of mH, with results in agreement with HDECAY[56,57].In Fig. 3 we plot � as a function ofmH, for � ¼ 45�. We

see that the interference effect is stronger for a heavierHiggs boson, and that it reaches �0:4% for mH ’150 GeV. This mass value is close to the region in whichthere may be sizable contributions to the phase from Wboson pairs, one on shell and one off shell in the H ! ��amplitude; so the plot cannot be extrapolated much furtherwithout performing this computation. In general, though,the dominant contribution to � for a light Higgs bosoncomes from the one-loop �� ! b �b and H ! b �bamplitudes.In Fig. 4 we plot � as a function of the scattering angle �,

for mH ¼ 130 GeV. Note that the small-mass approxima-tion formula (24) for � diverges for small angles. Thisbehavior can be understood as coming from the �� ! b �bcontinuum amplitude, which exhibits a similar angulardependence. Keeping the exact b-quark mass dependence,using Eq. (17), the divergence is regulated. We find that formH ¼ 130 GeV, � ¼ 18% at � ¼ 3�, and that it rolls off

-0.4

-0.3

-0.2

-0.1

0

90 100 110 120 130 140 150

δ (%

)

mH (GeV)

SM Higgs Interference Correction

θ = 45°all phases turned on

γ γ → H 1-loop phaseγ γ → b–b and H → b–b 1-loop phase

FIG. 3 (color online). The percentage reduction of the SMHiggs signal as a function of the Higgs boson mass, forcenter-of-mass scattering angle � ¼ 45�. The solid curve repre-sents the result with all phases turned on; the dashed curves turnon different component phases each time. The effect is strongerfor a higher mass Higgs boson.

LANCE J. DIXON AND YORGOS SOFIANATOS PHYSICAL REVIEW D 79, 033002 (2009)

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to a constant � � 35% for � < 0:5�. Of course it would bevery challenging experimentally to search for b jets in thisfar-forward region, and the reason � is increasing is be-cause the continuum b �b background is increasing. Awayfrom the forward region, the interference effect has theopposite sign, negative, and its magnitude becomes maxi-mal for � ’ 35�, with � ’ �0:18%. Again, the phase aris-ing from the one-loop �� ! b �b and H ! b �b amplitudesalmost solely determines the size of the correction.

In models beyond the SM, such as the MSSM, thecoupling of a Higgs boson to b quarks and to photons ismodified. How does the interference effect depend on thesecouplings? Looking at Eq. (24), we see that the two powersof the Yukawa coupling b � mb=v from jAtree

H!b �bj2 can-

cel against the ones contained in �H (which for most of therelevant range of mH is dominated by the H ! b �b decay).There is one extra power of b coming from the H ! b �bamplitude in the numerator in Eq. (5), so the dominantcontribution to � is linear in b. The subdominant contri-

bution from ImfAð1Þ��!Hg includes one more factor of b,

so it is quadratic in b.At a photon collider, the unperturbed peak height is

proportional to the product �ðH ! ��Þ � BrðH ! b �bÞ.The H ! �� width does not depend strongly on b untilit gets very large. The H ! b �b branching ratio is � 1,getting even closer to 1 as b increases. Thus, the unper-turbed peak height does not change dramatically, but thefractional shift � can increase considerably as b grows. In

particular for the MSSM, the Yukawa coupling to thelightest Higgs h is ðmb=vÞ � ðsin�= cos�Þ, where � is aHiggs mixing angle and the ratio of vacuum expectationvalues of Hu and Hd is tan�. If the heavier Higgs bosonsare not decoupled, and tan� is large (perhaps as large as�50), as in the so-called ‘‘intense coupling regime’’[58,59], then � can receive a big enhancement. As anexample, we have computed � assuming a factor of 20increase in b over the SM value; we obtain � � �4% formH ¼ 130 GeV and � ¼ 45�, with a significant contribu-

tion now from ImfAð1Þ��!Hg. (In the very-strong-coupling

regime one might also wish to compute corrections to �due to phases from rescattering via t-channel Higgs ex-change between the b quarks, but we have not done so.)From Eq. (24), � is inversely proportional to the H��

coupling, given by Eq. (18). This means that an enhance-

ment in � could also come from a decrease ofAð1Þ��!H, e.g.

by opposite-sign contributions from extra particles in theloop. However, such a decrease will also affect �ðH !��Þ, and consequently reduce the total number of events,leading to low statistics in the measurement of the Higgspartial width in the �� ! H ! b �b channel.In conclusion, we have presented results for the

resonance-continuum interference effect in the �� !H ! b �b channel at a photon collider, focusing on a low-mass (mH < 160 GeV) Higgs boson. We obtained ourresults by computing the relative phase arising from one-loop QCD corrections, exploiting the unitarity propertiesof the corresponding diagrams. We found that the domi-nant contribution comes from the one-loop �� ! b �b andH ! b �b amplitudes, and that the magnitude of the effect inthe SM is mostly within the range of 0.1%–0.2%. Thisindicates that such an interference effect is negligible forthe determination of the properties of the Higgs sector inthe SM, and probably negligible in most regions of MSSMparameter space, aside from ‘‘intense coupling’’ regions.The SM effect is an order of magnitude smaller than theexperimental precision achievable at a photon collider, andtherefore poses no worry for the measurement of the Higgspartial width at such a machine.

We would like to thank Michael Peskin for useful dis-cussions, and Jae Sik Lee, Apostolis Pilaftsis, and MichaelSpira for comments on the manuscript. The Feynman dia-grams in the paper were made with JaxoDraw [60], basedon AxoDraw [61]. This research was supported by theU.S. Department of Energy under Contract No. DE-AC02-76SF00515.

[1] P.W. Higgs, Phys. Rev. 145, 1156 (1966). [2] F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).

-0.2

-0.15

-0.1

-0.05

0

0.05

20 30 40 50 60 70 80 90

δ (%

)

θ (degrees)

SM Higgs Interference Correction

mH = 130 GeVall phases turned on

γ γ → H 1-loop phaseγ γ → b–b and H → b–b 1-loop phase

FIG. 4 (color online). The percentage reduction of the SMHiggs signal as a function of the scattering angle for mH ¼130 GeV. The solid curve represents the result with all phasesturned on; the dashed curves turn on different component phaseseach time. The total effect is maximized close to � ’ 35�.

RESONANCE-CONTINUUM INTERFERENCE IN LIGHT . . . PHYSICAL REVIEW D 79, 033002 (2009)

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