Electromagnetic Higgs production
arXiv:0704.1985v2  [hep-ph]  13 Nov 2007
Preprint typeset in JHEP style - HYPER VERSION December 8, 2021
Electromagnetic Higgs production
J. Miller
Department of Particle Physics, School of Physics and Astronomy
Raymond and Beverley Sackler Faculty of Exact Science
Tel Aviv University, Tel Aviv, 69978, Israel
Abstract: The cross section for central diffractive Higgs production is calculated, for the LHC range
of energies. The graphs for the possible mechanisms for Higgs production, through pomeron fusion and
photon fusions are calculated for all possibilities allowed by the standard model. The cross section for
central diffractive Higgs production through pomeron fusion, must be multiplied by a factor for the survival
probability, to isolate the Higgs signal and reduce the background. Due to the small value of the survival
probability
4 × 10−3
, the cross sections for central diffractive Higgs production, in the two cases for
pomeron fusion and photon fusion, are competitive.
Keywords: Higgs boson, BFKL pomeron, diffractive production, LHC, electromagnetic Higgs
production, photon photon fusion, pomeron pomeron fusion, survival probability, standard model.
Email: jeremymi@post.tau.ac.il;
http://arxiv.org/abs/0704.1985v2
http://jhep.sissa.it/stdsearch
http://jhep.sissa.it/stdsearch
Contents
1. Introduction 1
2. Double diffractive Higgs production at the LHC 4
3. Electromagnetic Higgs production at the LHC 8
3.1 The fermion triangle subprocess for Higgs production in γγ fusion 9
3.2 Boson loop sub-processes for Higgs production in γγ fusion 12
3.3 The cross section for central exclusive Higgs production through γγ fusion 15
4. Conclusion 15
5. Acknowledgements 15
A. Appendix 16
A-1 Evaluation of the integral over the anomalous dimensions γ1 and γ2 of the momentum Q
in the gluon density function 16
A-2 Feynman rules for the standard electro-weak theory 19
A-3 Evaluation of the integral over the momentum in the fermion triangle loop 19
1. Introduction
The most promising process for observation of the Higgs boson at the LHC is central diffractive production
of the Higgs, with large rapidity gaps ( LRG ) between the Higgs and the two emerging protons, after
scattering. Namely,
p + p → p + [LRG ] + H + [LRG ] + p (1.1)
The large rapidity gaps either side of the Higgs reduces the background, so that the Higgs signal will be
easier to isolate in central exclusive production. Hence, this process gives the best experimental signature
– 1 –
for detecting the Higgs at the LHC, and is very interesting for experiments in the search for the Higgs boson.
In this paper, two mechanisms are compared for central exclusive Higgs production; (1) γ γ fusion,
namely pp → γ γ → H and (2) pomeron exchange, namely pp → IP IP → H, where IP denotes a
pomeron. For γ γ fusion, there is no hard re-scattering of the photons to fill up the rapidity gaps, so that
large rapidity gaps are automatically present. The motivation for considering IP IP → H, is the observation
that for gluon gluon fusion, namely the process gg → H the colour flow induces many secondary parton
showers which fill up the rapidity gaps. Instead the process IP IP → H is a colour singlet exchange, where
the colour flow is screened, and the large rapidity gaps are preserved.
However in the case of IPIP fusion, there are hard re-scattering corrections, giving additional inelastic
scattering, which will give emission filling up the rapidity gaps (see ref. [1]). To guarantee the presence of
large rapidity gaps after scattering in central exclusive production, in the case of IPIP fusion, one has to
multiply by the survival probability. This is the probability that large rapidity gaps, between the Higgs
boson and the emerging protons, will be present after scattering.
The motive of this paper, was driven by the potentially small value for the survival probability, namely
< |S2| >= 4 × 10−3, calculated in ref. [1] for central exclusive Higgs production via IPIP fusion. This
is an order of magnitude less than previous estimates, namely < |S2| >= 0.02 for LHC energies (see ref.
[2]). Hence, previous calculations in ref. [2] of the cross section for central exclusive Higgs production via
IPIP fusion, which included a factor for the survival probability of < |S2| >= 0.02, gave for the exclusive
cross section σexcIP IP ( pp→ p+H + P ) = 3 fb. Since the survival probability is predicted in ref. [1] to be
an order of magnitude less, it follows that the cross section σexcIP IP ( pp→ p+H + P ) for central exclusive
Higgs production will also be an order of magnitude smaller. It also follows that this cross section will be
competitive with the cross section for central exclusive Higgs production via γ γ fusion, which is predicted
in ref. [2] to be σexcγγ ( p+ p→ p+H + p) = 0.1fb.
To illustrate that for the process of Eq. (1.1), the cross sections will be competitive for γγ fusion
and IPIP fusion, it is instructive to consider the diagrams for these processes. The notation used for the
couplings in the standard model are the following.
– 2 –
coupling constant expression value ( GeV / c2 )
1.17 × 10−5
v gw 80
2λ v2 120
) g2s(M2H)
where the mass of the Higgs boson, is derived from the vacuum expectation value of the Higgs SU (2)
weak isodoublet, which is
, where v =
, and µ and λ are parametres in the
Higgs potential, which is introduced into the standard model in the spontaneous symmetry breaking of
SUL ( 2) × UY ( 1) → UEM ( 1) , which is responsible for giving rise to the W and Z boson masses. In the
case for γγ fusion shown in Fig. 3, there are four vertices proportional to αem coupling the photons to the
two protons and the photons either side of the subprocess for γγ → H. A factor of g2w = 4
2GF m
should also be included, to account for the weak coupling of the Higgs to the sub-process, depicted by the
shaded area in Fig. 3. So it is expected that the cross section will be proportional to
σexcγγ ( p+ p→ p+H + P ) ∝ 4
2GF m
em = 0.6 fb (1.2)
where units are defined as 1GeV−2 = 0.3893 mb . In central exclusive Higgs production for the case
for γγ fusion, all the couplings of the photons shown in Fig. 3 are known constants, namely they are
proportional to αem. Hence, the cross section for this diagram can be calculated exactly. On the other
hand, when considering central exclusive Higgs production for the case for IPIP fusion shown in Fig. 1,
the couplings of the gluons are not constants. In Fig. 1 there are four gluon couplings with the protons,
giving a contribution to the cross section proportional to α4s
, where Q2 is the momentum transferred
along the pomeron, and it is assumed that αs
∼ 0.2. There are also two gluon couplings with the
subprocess for IPIP → H, giving a contribution to the cross section proportional to α2s
. Taking
the mass of the Higgs to be MH ∼ 100GeV , then it is expected that αs
∼ 0.12. Also in the case
of IPIP fusion, a factor of g2w = 4
2GF m
w should also be included, to account for the weak coupling
of the Higgs to the sub-process shown in Fig. 1. For IPIP fusion, this subprocess is the quark triangle
subprocess shown in Fig. 2. Since the gluon itself couples weakly to the Higgs boson, only the contribution
of the quark triangle subprocess is taken into account. The amplitude for the quark triangle subprocess, is
– 3 –
derived in section 3.1 for the electromagnetic case, where αs replaces αem. After multiplying by a factor
for the survival probability, which includes the survival probability < |S2
| >= 4× 10−3 which takes
into account hard re-scattering of the pomeron, and < |S2
| >= 5× 10−2 which takes into account soft
re-scattering of the pomeron, then one obtains for the exclusive cross section for central exclusive Higgs
production, in the case of IPIP fusion the value
σexcIP IP ( p+ p→ p+H + p) ∝ 4
2GF m
< |S2hard| >< |S
soft| >= 0.9 fb (1.3)
Comparing the estimates of Eq. (1.2) and Eq. (1.3), it is expected that σexcγγ ( p+ p→ p+H + p) and
σexcIP IP ( p+ p→ p+H + p) will be competitive. This is the motivation for this paper which re-examines
σexcγγ ( p+ p→ p+H + p) for electromagnetic Higgs production.
This paper is organised in the following way. In section 2, the details of the calculation of the cross
section for central exclusive Higgs production in the case for IPIP fusion is explained in detail. The cross
section which is obtained, is multiplied by the factor for the survival probability in ref. [1], to give the
exclusive cross section. In section 3, the cross section for central exclusive Higgs production, in the case
of γγ fusion is calculated. The mechanism γγ → H proceeds via the fermion triangle and Boson loop
sub-processes illustrated in Fig. 4 and Fig. 5. The total cross section for central exclusive Higgs production,
in the case of γγ fusion is calculated by taking the sum over all the contributions for these sub-processes
for the γγ → H mechanism. Finally, in the conclusion the results found in section 2 and section 3, are
compared.
All calculations in this paper, are based on the Feynman rules for the standard electro-weak theory,
given in Fig. 6, which is to be found in the appendix.
2. Double diffractive Higgs production at the LHC
In this section, an explicit expression is derived for the Born amplitude for double diffractive Higgs pro-
duction (see Fig. 1). The Higgs couples weakly to the gluon, so the main contribution comes from the
quark triangle subprocess (see Fig. 2), and all six flavours of quarks are taken into account. The second
t channel gluon in Fig. 1 is included. This is because of the large rapidity (LRG) gap between the Higgs
and the proton, which demands that in the t channel, one has colorless exchange. Indeed, if the LRG
wasn’t present between the protons, then the Higgs could simply be produced from gluon gluon fusion in
a single channel. However, the colour flow induced by a single channel exchange process, could produce
many secondary particles. These secondary particles could fill up the LRG. To screen the colour flow, it
is necessary to exchange a second t channel gluon. At lowest order in αs, this gluon couples only to the
incoming quark lines. The Born amplitude for double diffractive Higgs production by gluon exchange, is
given by the expression [3, 4]
– 4 –
Bremstrahlung
gluon
Higgs
t−channel
screening
gluon Q
BFKL gluons
BFKL gluons
Figure 1: Double diffractive Higgs production in the Born approximation
A~k1 · ~k2
~k1T · ~k2T
~k1T · ~k2T
(2.1)
where ~k1 ·~k2 =
has been used. In Eq. (2.1), the Weizsäcker - Williams approach, explained in ref.
[5] has been used. The factor A~k1 · ~k2 is the amplitude for the quark triangle subprocess of Fig. 2, where
A takes the value, [6, 7, 8, 9]
 (2.2)
The Born amplitude shown in Fig. 1, is extended to proton scattering instead of just quark scattering
here. The typical momentum transferred, t1 = ( q1 −Q) 2 and t2 = ( q2 −Q) 2, are rather small and of the
order of 1
, where b is the slope of the gluon - proton form factor, and can be estimated to be b = 5.5GeV 2.
Therefore, one takes into account the proton couplings to the gluon ladder (see Fig. 1), by including the
proton form factors in the gaussian form
bt1 −
where t1 = ( q1 −Q)2 and t2 = ( q2 −Q)2 (2.3)
– 5 –
Higgs
Figure 2: Quark triangle subprocess for Higgs production in gluon gluon fusion
to describe the dependence on the transferred momentum. If the momentum transfer is small, it can
be assumed that k1 ∼ k2 ∼ Q. Therefore, with the definition of Eq. (2.3), t1 → 0 and t2 → 0. Hence,
the proton form factors e−
bt1 and e−
bt2 tend to 1, and can be taken outside the integral, and the Born
amplitude behaves as
bt1e−
(2.4)
To consider the exclusive process only, with the condition of the LRG, bremsstrahlung gluons must be
suppressed. The bremsstrahlung gluons are shown by the dashed lines in Fig. 1, which are suppressed by
multiplying by the Sudakov form factor
Fs = e
−S(k2
) (2.5)
Fs is the probability not to emit bremsstrahlung gluons. S is the mean multiplicity of Bremsstrahlung
gluons given as
k2⊥, E
(2.6)
Secondly, evolution of BFKL ladder gluons between the two channels (see Fig. 1), must be taken into
account. For proton scattering instead of quark scattering, the naive coupling of the gluons to the external
– 6 –
quarks must be replaced by a coupling of the gluons to the external proton lines. To include both of these
modifications, the naive gluon density for quarks is replaced by the density for protons by the following
substitution
(2.7)
where f
x, k2
is the un-integrated gluon density of the proton. After including the Sudakov form
factor of Eq. (2.1), and the gluon density function of Eq. (2.7), the amplitude in Eq. (2.1) becomes
MIP IP (p+p → p+H+p) = Aπ3s
e−S(k
x1, Q
x2, Q
(2.8)
where for the gluon densities f (x1,2) = 2
γ1,2e
ω( γ1,2) ln
(2.9)
where γ1,2 are the anomalous dimensions and the numerical coefficient 2 in Eq. (2.9) can be taken
from MRST-2002-NLO parameterizations. Using Eq. (2.9), the integration over Q⊥ and over γ1 and γ2
can be evaluated. It turns out that the integrand of Eq. (2.8) has a saddle point given by ln Q2⊥ =
( γ1 + γ2 − 1) , and the essential values of γ1 and γ2 are close to 12 . Hence, the typical Q⊥ is
rather large, and depends on the mass of the Higgs. After integrating over γ1 and γ2 and q⊥, the final
result for the amplitude MIP IP ( p+p → p+H+p) is derived in the appendix (see section A-1), and the final
result is given in Eq. (A-1-16) as
MIP IP (p+p → p+H+p) =− 2Aπ4s
ω ”( 12) ln s1
ω ”( 12) ln s1
+ ω ”
ln s1
) (2.10)
Now that the amplitude is known, σIP IP ( p+p → p+H+p) for central Higgs production can be cal-
culated for the case of IPIP fusion. To derive the cross section for exclusive central Higgs production,
one has to multiply by a factor which takes into account the survival probability for large rapidity gaps
< |S2
| > = 0.004, to suppress hard re-scattering. Therefore, the cross section σexcIP IP ( p+p → p+H+p) ,
for central exclusive Higgs production, without any further hard re-scattering for the case of IPIP fusion,
takes the value
< |S2hard| > σIP IP ( p+p → p+H+p) = σ
IP IP ( p+p → p+H+p) = 0.47 fb (2.11)
– 7 –
3. Electromagnetic Higgs production at the LHC
In the case of central exclusive Higgs production for the case of γγ fusion, shown in Fig. 3, there is no hard
re-scattering to take into account, and all the couplings are known precisely. The shaded area in Fig. 3
depicts the subprocess for the mechanism γγ → H. The possible mechanisms are illustrated in Fig. 4 and
Fig. 5, and their contributions to the amplitude are calculated in this section. Gauge invariance requires
that the contribution of a subprocess for the mechanism γγ → H, takes the form
photon
photon
Higgs
+ qq=
Figure 3: Diffractive Higgs production in single channel photon exchange.
Aµν = A
(3.1)
where A is a constant, depending on the particular subprocess. It turns out that for the case of when
the subprocess is the fermion triangle shown in Fig. 4, summed over all six flavours of quarks, and all three
lepton flavours, then the expression for the amplitude takes the form of Eq. (3.1). However in the case of
when the subprocess is one of the Boson loops shown in Fig. 5, the expression for the amplitude is not of
the form of Eq. (3.1). The correct statement is that the sum over all the amplitudes for the sub-processes
shown in Fig. 5, gives a gauge invariant expression of the form of Eq. (3.1).
The amplitude for the diagram of Fig. 3, where the process γγ → H proceeds via the sum over the
fermion triangle shown in Fig. 4, and Boson loops shown in Fig. 5, is given by
Mγγ ( p+p → p+H+p) = −
4παem
Afµν +A
(3.2)
where A
µν = Af
denotes the amplitude of the quark/anti-quark triangle subpro-
cess, summed over all six quark flavours/anti-quark flavours (q, /q̄ = u/ū, d/d̄, s/s̄, c/c̄, t/t̄, b/b̄) and all
– 8 –
three lepton/anti-lepton flavours (L± = e±, µ±, τ±), shown in Fig. 4, and Abµν = Ab
denotes the amplitude of the sum over all the Boson loop sub-processes shown in Fig. 5. At this point, the
Weizsacker - Williams formula is used, as explained in refs. [3, 5, 4]. In this approach, the substitution
2Aµν = − 2SM2
2⊥Aµν is used. In the notation used in this paper, q
1⊥ and q
2⊥ denotes two dimen-
sional vectors, in the plane transverse to the direction of the momenta of the two incoming protons p
1 and
pν2 . Hence, the amplitude of Eq. (3.2) can be written as
Mγγ ( p+p → p+H+p) = −
4παem
Afµν +A
(3.3)
In order to calculate the cross section, one has to integrate the squared amplitude over all the transverse
momenta q1⊥ and q2⊥. For central exclusive Higgs production in the case of γγ fusion, it is required that
the lower limits of integration are
qmin1⊥
qmin2⊥
= m2p
, where mp is the proton mass, which is
assumed in this paper to be 1 GeV. The upper limits of integration, are taken from the electromagnetic
form factors for the proton, namely Gp
” , from which the upper limits of the integration
are derived to be ( qmax1⊥ )
= ( qmax1⊥ )
= 0.72.
σγγ ( p+p → p+H+p) =
(Af +Ab)
 (3.4)
where y = ln s
is the rapidity gap between the two incoming protons in Fig. 3, and m is the proton
mass, assumed to be 1 GeV. This calculation is for central exclusive Higgs production at the LHC, where
it is expected that
s = 14000GeV, which gives for the value of the rapidity gap y = 19.
3.1 The fermion triangle subprocess for Higgs production in γγ fusion
Central exclusive Higgs production for the case of γγ fusion, can proceed through the subprocess γγ →
fermion triangle → H shown in Fig. 4, where the fermions include the six flavours of quarks and anti
quarks (u , d , s , c , t , b) and the three lepton and anti lepton flavours (e , µ , τ ). A derivation of the
amplitude for the fermion triangle, can be found in refs. [6, 9, 10] for the case where the mass of the
fermion in the triangle is much larger than the Higgs mass. In this section, the amplitude of the fermion
triangle is derived by taking the sum over all the fermions which could contribute, including the six quark
flavours and the three lepton flavours. The H → f f vertex coupling, is proportional to the mass mf of
the fermion at the vertex, so that the sub-process amplitude of the fermion triangle will be proportional
to the mass of the fermion in the triangle. The fermion masses are assumed to take the following values
listed in the table below.
– 9 –
fermion mass ( GeV / c2 )
quarks
u 3 × 10−3
d 6 × 10−3
s 1.3
c 0.1
t 175
b 4.3
fermions
e 5.11 × 10−4
µ 0.106
τ 1.7771
Hence, these values indicate that the most significant contribution to the amplitude will come from
the top quark triangle.
Higgs
pH pH
Figure 4: Fermion triangle subprocess for Higgs production through γγ fusion
To derive explicitly the amplitude of the subprocess Fig. 4, the labeling of momenta shown in the
– 10 –
diagram is used, and the following notation is introduced.
D1 = l
2 −m2f D2 = (l − q1)
2 −m2f D3 = (l + q2)
2 −m2f (3.1.1)
where mf denotes the mass of the fermion which forms the triangle, which could be one of the quark
flavours or one of the lepton flavours. Then the amplitude for the fermion triangle subprocess, summed
over all possibilities of quark and anti-quark flavours, and lepton and anti-lepton flavours takes the form
Aµνq =
4παem
D1D2D3
where
L= e , µ , τ
(3.1.2)
where
q denotes the sum over all six quark flavours q = u , d , s , c , t , b and
L= e , µ , τ denotes
the sum over all three lepton flavours L = e , µ , τ . On the RHS of Eq. (3.1.2), d is the space-time
dimension, and at the end f the calculation, d → 4 is imposed. The reason for not specifying this in the
beginning, is because it will be necessary to use dimensional regularisation to cancel divergences, which
requires integration over d + ǫ dimensions, in the limit that d → 4 and ǫ → 0. Using the notation shown
in the diagram of Fig. 4 for the flow of momenta in the quark and anti - quark triangles, the trace term
is given by
lx + q
x +mf
lσ +mfγ
lτ − q1τ +mf
−lx + q1x +mf
−lσ +mfγµγτ
−lτ − q2τ +mf
= 8mf
2 + 4l
µlν + 2 (lµqν1 − lνq
2 )− g
~q1 · ~q2 + l2 −m2f
(3.1.3)
where ~q1 and ~q2 denotes four dimensional vectors. The first line on the RHS of Eq. (3.1.3) corresponds
to the contribution given by the triangle formed by the fermions, ( i.e. quarks q = (u , d , s , c , t , b )
and negatively charged leptons L− = ( e− , µ− , τ−) , and the second line corresponds to the contribution
given by the triangle formed by the anti - fermions ( i.e. anti - quarks q̄ = (ū , d̄ , s̄ , c̄ , t̄ , b̄ ) and posi-
tively charged leptons L+ = ( e+ , µ+ , τ+) ). Introducing Feynman parametres to rewrite the quotient
(D1D2D3)
in a more convenient form, Eq. (3.1.2) simplifies to
4παem
dd l̃
∫ 1−x
8mf I
l̃2 −∆f ( x, y)
where ∆f = m
f − M2H x y
and l̃µ = lµ − x qν1 + y q
2 (3.1.4)
Note in the last step, it was assumed that q21 ≪ q22 ≪ M2H ,m2f so that these terms can be ignored.
From the kinematics shown in the diagram of Fig. 3, and Fig. 4, it is clear that 2~q1 · ~q2 = M2H . The trace
– 11 –
term I
was given in Eq. (3.1.3) in terms of the unknown momentum lµ in the fermion triangle in Fig. 4.
In terms of the new variable l̃µ, the trace term takes the form
2 + 4l̃
µ l̃ν − 4qν1q
2x y − g
~q1 · ~q2 (1− 2x y)−m2f + l̃2
(3.1.5)
The details of the integration over the momentum l̃ on the RHS of Eq. (3.1.4) are given in section A-3
of the appendix, (see Eq. (A-3-1) - Eq. (A-3-4)). Here, dimensional regularisation is used, a technique
where one integrates over d + ǫ dimensions, and afterwards d → 4 and ǫ → 0. This removes non gauge
invariant terms, in the numerator of the integrand on the RHS of Eq. (3.1.4). In this way, one obtains the
following gauge invariant result for the RHS of Eq. (3.1.4).
2αemG
If (3.1.6)
where
f =u , d , s , c , t , b , e , µ , τ
∫ 1−x
1− 4xy
∆f ( x, y)
where ∆f ( x, y) = m
f −M2Hxy (3.1.7)
The integral If is evaluated in section A-3 of the appendix (see Eq. (A-3-5) - Eq. (A-3-13) ). It turns
out that, due to the dependence of the factor for the H → f f vertex coupling on the mass of the fermion
mf , that the only fermion triangle which gives a significant contribution to the amplitude is for the case
where mf ≫ MH . From the table, this is true only for the top quark mt. Hence, it turns out that the
only fermion triangle that is necessary to take into, is the top quark triangle, and the contributions from
the rest of the triangle sub-processes formed by the rest of the quarks, and the leptons can be neglected.
Using this result, the amplitude for the contribution of the fermion triangle has the expression
= Af (q
2 − g
µν(~q1 · ~q2)) where Af = −
(3.1.8)
3.2 Boson loop sub-processes for Higgs production in γγ fusion
For central exclusive Higgs production, the Higgs can also be produced through the subprocess γγ →
boson loop → H, where the possible boson loops are shown in Fig. 5 ( taken from ref. [7]). H− in Fig. 5
is an un-physical, charged Higgs boson, and φ± is a Fadeev-Popov ghost. The formalism for calculating
the amplitude of each diagram in Fig. 5, is similar to the approach used to calculate the amplitude of the
fermion triangle above, in section 3.1. Similarly here, after integration over the unknown momentum l in
the loop of each diagram in Fig. 5, and after integration over Feynman parametres, the expression for each
diagram takes the general form [7]
– 12 –
gµν + C
(3.2.1)
where d is the dimension of space-time. Terms proportional to q21 and q
2 were assumed to vanish, and
from the kinematics shown in Fig. 3, it was assumed that ~q1 · ~q2 =
. In the limit that d → 4, the
term proportional to Γ
on the RHS of Eq. (3.2.1) tends to infinity. However, when one sums over
the contributions to the amplitude given by all the boson sub-processes shown in Fig. 5, these divergencies
cancel exactly (see table below). One requires also, that this sum over all the amplitudes for the boson
loop sub-processes shown in Fig. 5, satisfies the condition
C = −
D (3.2.2)
such that the amplitude for the sum is gauge invariant. In ref. [7], this sum was taken and the result was
an almost gauge invariant expression, since terms proportional to
and higher were neglected. In the
calculation which lead to the results in this paper, the result gives an exactly gauge invariant expression,
after using dimensional regularisation to remove terms which do not satisfy the gauge invariance condition.
graph B C D
a + crossed 3 ( d− 1) −4 5
b + crossed −2 ( d− 1) 0 2m
c + d + crossed −1
( d− 1) −4 2
e + crossed 0 0 −m
f + crossed 0 0 1
g + h + crossed 1 0 0
i + crossed −2 0 0
2j + crossed −1
sum 1
d − 2 −8 8
Hence, plugging the results shown in the table for the coefficients into Eq. (3.2.1), the result of taking
the sum over the amplitudes for the sub-processes shown in Fig. 5 gives a gauge invariant expression, and
in the limit that d→ 4, the divergencies cancel exactly, such that the expression of Eq. (3.2.1) reduces to
where Ab = 4
(3.2.3)
– 13 –
Higgs
Figure 5: Boson loop subprocess for Higgs production in γγ fusion
It should be noted from the results for the amplitude of the subprocess of Fig. 5 (a), the subprocess
γγ → W triangle → H, interferes destructively with the subprocess γγ → fermion triangle → H shown
in Fig. 4.
– 14 –
3.3 The cross section for central exclusive Higgs production through γγ fusion
Now that the amplitudes for the sub-processes γγ → fermion triangle → H and γγ → boson loop → H
have been calculated, the results can be plugged into Eq. (3.4) to derive the cross section, for central
exclusive Higgs production through γγ fusion. The result, taking into account all possible sub-processes
shown in Fig. 4 and Fig. 5 is found to be
σexcγγ ( p+p → p+H+p) = 0.1 fb (3.3.1)
4. Conclusion
The results of this paper are summarized in the table below. σexc is the exclusive cross section, which
includes multiplication by a factor for the survival probability, for central exclusive Higgs production. The
results are given for the mechanisms pp → γγ → H and pp → IPIP → H. Note that in these results, the
cross section for central exclusive Higgs production in the case of γγ fusion, is multiplied by a factor for the
survival probability of 1. This is because in the case of photon exchange, there is no hard re-scattering to
suppress, and the large rapidity gaps between the Higgs and the two emerging protons are automatically
present.
process < |S2 | > σexc (fb)
IPIP 0.023 2.7
IPIP 0.004 0.47
γγ 1 0.1
The results show that, taking the survival probability to be 0.02, which is the value used in ref. [2], then
the result for σexcIP IP for central exclusive Higgs production at the LHC, almost agrees with the prediction of
ref. [2], (which was 3 fb). However, if the survival probability is an order of magnitude smaller as predicted
in ref. [1], then σexcIP IP will be an order of magnitude smaller and, it becomes competitive with σ
γγ for
central exclusive Higgs production at the LHC.
5. Acknowledgements
This paper is dedicated to the memory of Grandpa Herman, the Pindenjara April 15th 1924 - January 2nd
2007. I would like to thank E. Levin for helpful advice in writing this paper. I would also like to thank E.
Gotsman, A. Kormilitzin, A. Prygarin for fruitful discussions on the subject. This research was supported
in part by the Israel Science Foundation, founded by the Israeli Academy of Science and Humanities, by a
grant from the Israeli ministry of science, culture & sport & the Russian Foundation for Basic research of
the Russian Federation, and by the BSF grant 20004019.
– 15 –
A. Appendix
A-1 Evaluation of the integral over the anomalous dimensions γ1 and γ2 of the momentum
Q2 in the gluon density function
The Born amplitude was calculated in Eq. (2.8), in terms of the gluon density as a function of the anomalous
dimensions γ1 and γ2, for the two gluon ladders in Fig. 1. One now needs to integrate over γ1 and γ2, and
also over the momentum in the t-channel gluon, namely Q. Altogether the necessary integrations take the
MIP IP ( p+p → p+H+p) = Aπ3s
dγ1dγ2
e−S(k
x1, Q
x2, Q
(A-1-1)
Firstly, the integral over Q⊥ is evaluated using the steepest descent technique. The gluon density is
given by
Q2, x1,2
)γ1,2
ω(γ1,2) ln
s0 (A-1-2)
where ln
, where s0 ∼ 1GeV . This comes from the BFKL ladder gluon exchange
(see Fig. 1), while the coefficient was taken from MRST - NLO - 2002 data (see Ref/[?]). ω (γ1,2) is the
BFKL kernel defined as
ω (γ1,2) = ᾱsχ (γ1,2) = ᾱs (ψ ( 1) − ψ ( γ1,2) − ψ ( 1− γ1,2) ) (A-1-3)
where ψ ( f) is the digamma function and ψ ( f) =
dΓ( f)
. In Eq. (A-1-2), S
k2⊥, E
is the Sudakov
form factor with the typical value [3, 4] S
Q2⊥, E⊥
= 3αs
, in the notation that E⊥ =
Using this substitution Eq. (A-1-1) then becomes
MIP IP ( p+p → p+H+p) = 4Aπ3s
dγ1dγ2
× exp
ω ( γ1) ln
+ ω ( γ2) ln
(A-1-4)
where φ
− (γ1 + γ2 − 1) lnQ2⊥ (A-1-5)
Differentiating the right hand side of Eq. (A-1-5) with respect to ln Q2⊥, one sees that φ has a saddle
point at ln Q2⊥ = ln
(γ1 + γ2 − 1). Hence, changing the integration variable to u = ln Q2⊥, and
expanding φ around the saddle point, Eq. (A-1-4) can be written as
– 16 –
MIP IP ( p+p → p+H+p) = 4Aπ3se−φ(u0)
dγ1dγ2
(u−u0)2
d2φ(u0)
ω( γ1) ln
+ω( γ2) ln
(A-1-6)
where u0 = ln
(γ1 + γ2 − 1) (A-1-7)
Now the right hand side of Eq. (A-1-6) has reduced to a Gaussian integral over u, which can be
evaluated by the steepest descent technique, to give the expression
MIP IP ( p+p → p+H+p) =4Aπ4s
dγ1dγ2 exp
(γ1 + γ2 − 1)
(γ1 + γ2 − 1) + ln
× exp
ω ( γ1) ln
+ ω ( γ2) ln
(A-1-8)
The BFKL function ω ( γ) has a saddle point at γ = 1
. Near to this point ω ( γ) can be written as
ω (γ1,2) = ω
γ1,2 −
(A-1-9)
Hence using Eq. (A-1-9), Eq. (A-1-8) can be reduced to
MIP IP ( p+p → p+H+p) =4Aπ4s
dγ1dγ2 exp ( f ( γ1, γ2) ) (A-1-10)
where the function f ( γ1, γ2) has the form
f (γ1, γ2) =ω
+ (γ1 + γ2 − 1)
(γ1 + γ2 − 1) + ln
(A-1-11)
This function has a saddle point with respect to γ1 given by
(γ2 − 1)− ln
ln s1
+ ω ”
ln s1
) (A-1-12)
Hence, expanding f ( γ1, γ2) around γ
1 , the integration over γ1 is evaluated using the steepest descent
technique to give the expression
– 17 –
MIP IP ( p+p → p+H+p) =2Aπ4s
exp ( f (γ
1 , γ2))
+ ω ”
ln s1
(A-1-13)
Now the function f (γ
1 , γ2) has a saddle point with respect to γ2 given by
ln s1
ln s1s2
+ ω ”
ln s1s2
1 + 4π
ln s1
(A-1-14)
Here in the second line it is assumed that s1 ∼ s2. For large s1 and s2, γsp2 is approximately
. Using
the same method as above, expanding f ( γ
1 , γ2) around γ
2 , the integration over γ2 is evaluated using
the steepest descent technique for Eq. (A-1-13), to give the result
MIP IP ( p+p → p+H+p) = Aπ4s
2 −2 exp
1 , γ
2 ∼ 12
+ ω ”
ln s1
) (A-1-15)
where f ( γ
1 , γ
2 ) = −
ln s1
ln s1
(A-1-16)
– 18 –
A-2 Feynman rules for the standard electro-weak theory
wmwgµν
w w H
νρ µ ρµ ν µν ρ
µνgρσ µρgνσ µσgνρ
Figure 6:
Figure 7: Feynman rules in the standard electroweak theory
A-3 Evaluation of the integral over the momentum in the fermion triangle loop
The amplitude for the fermion triangle subprocess, summed over all quark flavours q = (u , d , s , c , t , b )
– 19 –
and lepton flavours L = ( e , µ , τ ) , for the mechanism γγ fermion triangle → H was found in equation
Eq. (3.1.4) to take the form
4παem
∫ 1−x
( 1− 2x y) − 2x y M
gµν +m2
l̃2 −∆f (x, y)
4παem
∫ 1−x
4l̃µ l̃ν − l̃2 gµν
l̃2 −∆f ( x, y)
where
∆f = m
f − M2H xy and
L= e , µ , τ
(A-3-1)
where
q denotes the sum over all six quark flavours q = u , d , s , c , t , b and
L= e , µ , τ denotes the
sum over all three lepton flavours L = e , µ , τ .
From the numerator of the integrand, on the RHS of Eq. (A-3-1), one can see that there is a gauge
invariant term
( 1− 4x y) , and the numerator in the integrand of the second line on
the RHS gives a vanishing contribution to the integration over l̃, for d → 4. However one is still left
with the terms −2xyM
gµν + m2
gµν in the numerator of the integrand, which are certainly not gauge
invariant. However, conveniently this non gauge invariant piece is exactly equal to ∆f (x, y) introduced
in Eq. (3.1.2). To deal with this non gauge invariant piece, it is useful to use dimensional regularisation,
when integrating over the l̃2 term in the numerator of the integrand on the second line. In this approach,
one initially integrates over d+ ǫ dimensions in the limit that ǫ→ 0 and d→ 4. In this way the non gauge
invariant terms disappear. Hence, evaluating the integral over l̃ on the RHS of Eq. (A-3-1) gives
4παemG
(−1)3 Γ
3 − d
Γ ( 3) ( 4π)
∫ 1−x
( 1− 4x y) + ∆f (x, y) gµν
∆f (x, y)
+ 2 lim
4παemG
(−1)3+1 Γ
3 − d+ǫ
2Γ ( 3) ( 4π)
∫ 1−x
dy 8m2f ( 4g
µν − ( d+ ǫ) gµν) (A-3-2)
In the limit that ǫ → 0 and d → 4, the gamma function Γ
3− d+ǫ
and the RHS of
Eq. (A-3-2) reduces to
4παemG
2 ( 4π)
∫ 1−x
( 1− 4x y) + ∆f (x, y) gµν
∆f ( x, y)
4παemG
2 ( 4π)
∫ 1−x
dy 8m2f ǫ g
µν (A-3-3)
– 20 –
Thus, after canceling ǫ in the numerator and the denominator in the second line on the RHS of Eq. (A-
3-3), the second line exactly cancels the non gauge invariant part of the integrand on the first line. Hence,
one is left with the purely gauge invariant expression
Aµνq = −2
If (A-3-4)
where If is the only remaining integral to evaluate, which takes the form
f =u , d , s , c , t , b ,e , µ , τ
∫ 1−x
1− 4xy
∆f (x, y)
where ∆f (x, y) = m
f −M2Hxy
(A-3-5)
To evaluate this integral, this are two cases to consider, namely (1) when m2
≫ M2H , which is true
for the top quark when mf = mt = 175GeV, and (2) when m
≪ M2H , which is true for all the rest of
the fermions listed in the table. Therefore
f If can be separated into two parts, namely
If = It +
f 6= t
If (A-3-6)
For case (1), where mf = mt ≫ MH , one can write It in a more convenient way as
∫ 1−x
1− 4xy
∫ 1−x
 4 +
1− 4 m
) ∫ 1
x ( 1− x)
) ∫ 1
( 1− x) +
x ( 1− x)2 +
x2 ( 1− x)3 + .....
(A-3-7)
where in the last step, the logarithm was expanded in a Taylor series around x = 0. Evaluating the
integral over x, and since it is assumed that M2H ≪ m2t , retaining terms no smaller than
, the RHS
of Eq. (A-3-7) becomes
+ ....
(A-3-8)
– 21 –
For case (2), where mf ≪ MH which includes all the fermions in the table except for the top quark,
there are two possible regions of integration, namely (I) when M2H x y > m
and (II) when M2H x y < m
In the region where M2H x y > m
f , the RHS of Eq. (A-3-5) reduces to
f 6= t
region (I)
f 6= t
∫ 1−x
1− 4xy
f 6= t
x ( 1− x)
f 6= t
1− x −
f 6= t
polylog ( 2 , x = 1) −
f 6= t
polylog
2 , x =
f 6= t
 (A-3-9)
In the region (II) where M2H x y < m
, the RHS of Eq. (A-3-5) reduces to
f 6= t
region (II)
f 6= t
( 1− 4xy)
f 6= t
f 6= t
(A-3-10)
Hence, adding the contributions of Eq. (A-3-9) and Eq. (A-3-10) for the contributions of region (I) and
region (II) of the integral, gives the result
f 6= t
If = −
f 6= t
f 6= t
polylog ( 2 , x = 1) − polylog
2 , x =
(A-3-11)
f 6= t
for mf ≪ MH (A-3-12)
From Eq. (A-3-12) and Eq. (A-3-8), the result for the evaluation of the integral If has its main
contribution from the top quark triangle, such that
– 22 –
If ≈ It , =
for mt ≫ MH (A-3-13)
Plugging this result into Eq. (A-3-4) gives the final expression for the amplitude of the fermion triangle
subprocess shown in Fig. 4, for the sum over all quark q = ( u , d , s , c , t , b ) contributions and lepton
contributions L = e , µ , τ as
where Af = −
(A-3-14)
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