Can the diphoton enhancement at 750 GeV be due to a neutral technipion?

# Can the diphoton enhancement at 750 GeV be due to a neutral technipion?

## Abstract

We discuss a scenario in which the diphoton enhancement at = 750 GeV, observed by the ATLAS and CMS Collaborations, is a neutral technipion . We consider two distinct minimal models for the dynamical electroweak symmetry breaking. In a first one, two-flavor vector-like technicolor (VTC) model, we assume that the two-photon fusion is a dominant production mechanism. We include and production of technipion associated with one or two jets. All the considered mechanisms give similar contributions. With the strong Yukawa (technipion-techniquark) coupling = 10 - 20 we obtain the measured cross section of the “signal”. With such values of we get a relatively small . In a second approach, one-family walking technicolor (WTC) model, the isoscalar technipion is produced dominantly via the gluon-gluon fusion. We also discuss the size of the signal at lower energies (LHC, Tevatron) for (VTC) and jet-jet (WTC) final states and check consistency with the existing experimental data. We predict a measurable cross section for production associated with one or two soft jets. The technipion signal in both models is compared with the SM background diphoton contributions. We observe the dominance of inelastic-inelastic processes for induced processes. In the VTC scenario, we predict the signal cross section for purely exclusive processes at = 13 TeV to be about 0.2 fb. Such a cross section would be, however, difficult to measure with the planned integrated luminosity. In all considered cases the signal is below the background or/and below the threshold set by statistics.

###### pacs:
14.80.Ec, 14.80.Bn, 12.60.Nz, 14.80.Tt, 12.60.Fr

LU TP 16-XX

March 2016

## I Introduction

Recently both the ATLAS and CMS Collaborations announced an observation of an intriguing enhancement in the diphoton invariant mass at  GeV in proton-proton collisions at  TeV ATLAS:2015 (); CMS:2015dxe () 2. Remarkably, such a hint to a possible New Physics signal has triggered a lot of research activities in recent months looking for its possible interpretation in various theoretical scenarios of New Physics (see e.g. Refs Ellis:2015oso (); Franceschini:2015kwy (); Csaki:2015vek (); Fichet:2015vvy (); Fichet:2016pvq ()). However, before one could be certain about a possible nature of such a hint, it requires further confirmation by collecting a better statistics. If it is confirmed it will be a very important discovery related to first observation of the signal beyond Standard Model (SM). Different scenarios are possible a priori. The resonance signal observed means that the potentially new state decays into (and thus should couple to) two photons. What is the dominant production mechanism is a speculation at this stage. Several options are possible a priori. In one of them gluon-gluon fusion is the dominant production mechanism. If coupling of the new state to gluons is weak other options have to be considered. In the present analysis we consider such an example. The two-photon induced production of various objects became recently rather topical. This includes, for instance, production of Luszczak:2011uh (), Maciula:2010vc (); Maciula:2010tv (), daSilveira:2014jla (); Luszczak:2015aoa (), Lebiedowicz:2012gg (); Luszczak:2014mta () or Lebiedowicz:2015cea (). In the current study, we continue this line of research and consider the two-photon production mechanism of a new lightest composite state, the technipion predicted by various technicolor models, thus probing its potential to explain the 750 GeV excess.

A high-scale strongly-coupled physics can, in principle, be responsible for the dynamical electro-weak symmetry breaking (EWSB) in the SM by means of strongly-interacting technifermion condensation (see e.g. Refs Weinberg:1975gm (); Susskind:1978ms (); Eichten:1979ah ()). Such a dynamics typically predicts a plenty of new composite states close to the EWSB scale, in particular, relatively light composite scalar Higgs-like particles and pseudoscalar technipions. A consistent realisation of the underlined compositeness scenarios is typically limited by the precision SM tests Peskin:1990zt (); Peskin:1991sw (); Galloway:2010bp () and the ongoing SM-like Higgs boson studies Arbey:2015exa () (for a detailed review see e.g. Refs Hill:2002ap (); Sannino:2009za ()). One of the appealing and consistent classes of TC models with a vector-like (Dirac) UV completion is known as the vector-like TC (VTC) scenario Kilic:2009mi (). The simplest version of the VTC scenario applied to the EWSB possessed two Dirac techniflavors and a SM-like Higgs boson Pasechnik:2013bxa (); Lebiedowicz:2013fta (); Pasechnik:2014ida (). Recently, the concept of Dirac UV completion has also emerged in composite Higgs boson scenarios with confined symmetry Cacciapaglia:2014uja (); Hietanen:2014xca ().

Below, we discuss possible implications of the neutral pseudoscalar technipion in the two-flavor VTC Pasechnik:2013bxa () and one-family Walking TC (WTC) Matsuzaki:2015che () scenarios of the dynamical EWSB for the diphoton 750 GeV signature at the LHC. In the simplest VTC scenario, the technipion does not couple to quarks and gluons and is thus produced only via EW vector ( and ) boson fusion (VBF) mechanism with a slight dominance of the fusion channel at large invariant masses. The corresponding technipion production processes can be classified into three groups depending on the QED order. The first group of diagrams is shown in Fig. 1. Diagrams in Fig. 2 show the higher QED-order group, that is, the technipion production associated with one jet. We consider here the and subprocesses. In even higher QED-order we have to include also subprocesses, where and can be either a quark or an antiquark of various flavours from each of the colliding protons. In the Walking TC scenario, the technipion is produced dominantly by the ordinary gluon-gluon fusion (relevant also for the Higgs boson production at the LHC) and can decay with sizeable branching fraction into two-photon final state3.

## Ii Vector-like TC model: a short overview

Before we go to the production mechanisms relevant for the neutral technipion production, let us summarize the main points of the VTC model.

The Dirac techniquarks in the minimal two-flavor VTC model form pseudoreal representations of the global group which contains the chiral symmetry in the technimeson sector. In the linear realisation of the VTC model, at low energies the global chiral symmetry describes the effective interactions the lightest technimeson states, technipion (), technisigma and constituent Dirac techniquarks Pasechnik:2013bxa (); Pasechnik:2014ida (), similar to those in quark-meson effective theories of QCD hadron physics Tetradis:2003qa ().

For the Dirac UV completion, the vector subgroup of the global chiral group and the local weak-isospin symmetry of the SM are locally isomorphic. Therefore, the representations of the can always be mapped onto the representations of such that the , and Dirac can be classified as triplet, singlet and doublet representations of the gauge symmetry. Such a straightforward way of introducing weak interactions into the technimeson sector was proposed in the framework of the VTC model in Ref. Pasechnik:2013bxa (). In particular, it was demonstrated for the first time that practically any simple Dirac UV completion with chirally-symmetric weak interactions naturally escapes the electroweak precision constraints which is considered the basic motivation for the VTC scenario despite its simplicity.

Let us consider a single doublet of Dirac techniquarks

 ~Q=(UD), (1)

confined under at an energy scale above the EW scale. For a QCD-like scenario we choose and the hypercharge provided that electric charges of corresponding bounds states are integer-valued. Then, the phenomenological interactions of the constituent techniquarks and the lightest technimesons are described by the (global) chiral invariant low-energy effective Lagrangian in the linear -model (LM)

where the “source” term linear in technisigma is proportional to the flavor-diagonal techniquark condensate , , , and the corresponding covariant derivatives read

 ^D~Q=γμ(∂μ−iY~Q2g′Bμ−i2gWaμτa)~Q, DμPa=∂μPa+gϵabcWbμPc. (3)

For simplicity, the Higgs boson doublet is kept to be elementary. The choice of the “source” term in Eq. (2) is rather natural since it (a) induces a pseudo-Goldstone mass scale for pseudo-Goldstone technipion , and (b) relates the scales of the spontaneous EW and chiral symmetry breakings as well as the constituent techniquark mass scale with the value of the techniquark condensate Tetradis:2003qa (); Pasechnik:2013bxa ().

In the conformal limit of the theory , the EW and chiral symmetries are broken by the Higgs and technisigma vevs

 H=1√2(√2iϕ−H+iϕ0),⟨H⟩≡v,⟨S⟩≡u≳v, H=v+hcθ−~σsθ,S=u+hsθ+~σcθ, (4)

respectively, which are initiated by the techniquark condensation in the confined regime, i.e.

 u=(gTCλHδ)1/3|⟨¯~Q~Q⟩|1/3, v=(|λ|λH)1/2(gTCλHδ)1/3|⟨¯~Q~Q⟩|1/3. (5)

Then, technipions and techniquarks acquire a dynamical effective mass

 m2π=−gTC⟨¯~Q~Q⟩u, mU=mD≡m~Q=gTCu.

Above, , , , and . The phenomenologically consistent regime corresponds to a small Higgs-technisigma mixing which is realised in the TC decoupling limit Pasechnik:2013bxa (). As a characteristic feature of the model, the technipions in the VTC model can stay relatively light and do not have tree-level couplings to the SM fermions, so can only be produced in vector-boson fusion channels Pasechnik:2013bxa (); Lebiedowicz:2013fta (). In what follows, we discuss possible signatures of the VTC technipions at the LHC.

Decay widths for the neutral technipion in the vector-like TC model

The techniquark-loop amplitude has the following form Pasechnik:2013bxa ()

 iV~π0V1V2=FV1V2(m21,m22,m2~π0;m2~Q)ϵμνρσpμ1pν2ε∗1ρε∗2σ, (6) FV1V2=NTC2π2∑~Q=U,Dg~QV1g~QV2g~Q~π0m~QC0(m21,m22,m2~π;m2~Q), (7)

where is the standard finite three-point function, , and are the four-momenta, polarization vectors of the vector bosons and their on-shell masses, respectively, and neutral technipion couplings to techniquarks are

 gU~π0=gTC,gD~π0=−gTC, (8)

while gauge couplings of techniquarks are defined in Ref. Pasechnik:2013bxa (). Finally, the explicit expressions of the effective neutral technipion couplings for on-shell , and final states are4

 Fγγ=4αemgTCπm~Qm2~π0arcsin2(m~π02m~Q),m~π02m~Q<1, (9) FγZ=4αemgTCπm~Qm2~π0cot2θW[arcsin2(m~π02m~Q)−arcsin2(mZ2m~Q)], (10) FZZ=2αemgTCπm~QC0(m2Z,m2Z,m2~π0;m2~Q), (11)

where is the fine structure constant.

Now the two-body technipion decay width in a vector boson channel can be represented in terms of the effective couplings (7) as follows:

 Γ(~π0→V1V2)=rVm3~π64π¯λ3(m1,m2;m~π)|FV1V2|2, (12)

where for identical bosons and and for different ones, and is the normalized Källén function

 ¯λ(ma,mb;q)=(1−2m2a+m2bq2+(m2a−m2b)2q4)1/2. (13)

For example, in the VTC model, for = 10 and one gets:

 Γ(~π0→γγ) = 5.136×10−3GeV, (14) Γ(~π0→γZ) = 4.376×10−3GeV, (15) Γ(~π0→ZZ) = 4.734×10−3GeV. (16)

The total decay width is a sum of the tree contributions:

 Γtot=Γ(~π0→γγ)+Γ(~π0→γZ)+Γ(~π0→ZZ). (17)

The corresponding branching fractions are:

 Br(~π0→γγ) = 0.36, (18) Br(~π0→γZ) = 0.31, (19) Br(~π0→ZZ) = 0.33. (20)

How the total decay width depends on the model coupling constant is shown in Fig. 4. Only at very large = 300-400 one can reproduce the ATLAS quasi-experimental value 45 GeV ATLAS:2015 (); CMS:2015dxe () (only the ATLAS collaboration claims that the observed state is broad). Such a gigantic value is far too big compared to an analogical coupling in effective quark-hadron interactions in low-energy QCD such that the model looses its physical sense. However, the extraction of the total width from the “experimental data” is highly speculative and biased and thus should not be taken too seriously.

## Iii Production mechanisms of neutral technipion

As mentioned in the introduction we shall consider contributions of different QED-orders to the production of hypothetical technipions as shown in Figs. 1-3. Let us briefly discuss all the contributions one by one.

As already discussed above in our VTC model Pasechnik:2013bxa () the technipion couples only to photons and bosons. In the present paper, we shall include only coupling to photons as far as the production mechanism of technipion is considered. The couplings to bosons do not affect the observables significantly and will be considered elsewhere.

### iii.1 2→1 subprocess

The corresponding diagrams in Fig. 1 can be categorized into three groups (elastic-elastic, elastic-inelastic, inelastic-inelastic) depending whether the protons survive intact or undergo electromagnetic dissociation.

The cross section for the contribution via the subprocess can be easily written in the compact form:

 dσpp→~π0dy~π0=πm4~π0∑i,jx1γ(i)(x1,μ2F)x2γ(j)(x2,,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mγγ→~π0|2, (21)

where indices and denote el or in, i.e. they correspond to elastic or inelastic flux (-distribution) of equivalent photons, respectively. The elastic photon flux can be calculated using e.g. Drees-Zeppenfeld parametrization Drees:1994zx (). The factorization scale makes sense only in the case of dissociation of a proton. Here we take . Above is rapidity of the technipion and

 x1=m~π0√sexp(y~π0),x2=m~π0√sexp(−y~π0). (22)

In the leading-order collinear approximation the technipion is produced with zero transverse momentum. To calculate inelastic contributions we use collinear approach with photon PDFs Martin:2004dh ().

The matrix element squared has been calculated with effective vertex. In general, the form factor (9) describes the coupling of two photons to the technipion resonance could depend on virtualities of photons.

### iii.2 2→2 subprocess

Now we discuss diagrams shown in Fig. 2 with the and subprocesses. The cross section can be also written in a compact way which allows easy calculation of differential distributions:

 dσdy3dy4d2pt,~π0 = 116π2^s2∑ix1γ(i)(x1,μ2F)x2qeff(x2,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mγq→~π0q|2, (23) + 116π2^s2∑jx1qeff(x1,μ2F)x2γ(j)(x2,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mqγ→~π0q|2,

where index “3” refers to technipion and index “4” refers to outgoing quark/antiquark and

 x1=m1⊥√sexp(y3)+m2⊥√sexp(y4),x2=m1⊥√sexp(−y3)+m2⊥√sexp(−y4), m1⊥=√m2~π0+p2t,~π0,m2⊥=pt,~π0. (24)

In this approach, transverse momenta of and outgoing are strictly balanced. Here we have introduced effective parton distribution which we define as:

 qeff(x,μ2)=∑fe2f(qf(x,μ2)+¯qf(x,μ2)), (25)

where we take .

The matrix element for the process including masses of quarks reads:

 Mγq→~π0q=Fγγεμνκαp1μp3νε(γ)κ(p1,λ1)−igαβk2¯u(p4,λ4)γβu(p2,λ2), (26)

were . The matrix element squared can be written in terms of the Mandelstam variables as

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mγq→~π0q|2= 14F2γγ2e2^t2[2(^s−m2q)(k⋅p1)(k⋅p4)+2(m2q−^u)(k⋅p1)(k⋅p2) (27) −4m2q(k⋅p1)2+^t(m2q−^s)(m2q−^u)], ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mqγ→~π0q|2= 14F2γγ2e2^t2[2(^s−m2q)(k⋅p2)(k⋅p4)+2(m2q−^u)(k⋅p2)(k⋅p1) (28) −4m2q(k⋅p2)2+^t(m2q−^s)(m2q−^u)].

### iii.3 2→3 subprocess

As the last contribution we discuss the diagram in Fig. 3. The cross section for the partonic process can be written as:

 σqq′→q~π0q′=12^s¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mqq′→q~π0q′|2Jdξ1dξ2dy~π0dϕ12, (29)

where is the relative azimuthal angle between and , and , where and are transverse momenta of outgoing and , respectively.

The matrix element for the subprocess was calculated as:

 Mqq′→q~π0q′(λ1,λ2,λ3,λ4)=e2¯u(p3,λ3)γμu(p1,λ1)−igμν^t1 ×ενν′αβq1αq2βFγγ−igν′μ′^t2¯u(p4,λ4)γμ′u(p2,λ2). (30)

For comparison, we shall also calculate the matrix element in the high-energy approximation:

 ¯u(p′,λ′)γμu(p,λ)→(p′+p)μδλ′λ, (31)

often used in the literature in different context, see e.g. LABEL:Lebiedowicz:2015cea. We have also obtained a formula for matrix element squared and checked that it gives the same result as the calculation with explicit use of spinors.

The total (phase-space integrated) cross section for technipion production could be alternatively calculated as:

 σpp→~π0jj=∫dx1dx2∑f1,f2qf1(x1,μ2F)qf2(x2,μ2F)^σf1f2→f1~π0f2(^s). (32)

Limiting to -fusion processes only one can write:

 σγγpp→~π0jj=∫dx1dx2qeff(x1,μ2F)qeff(x2,μ2F)^σeffqq→q~π0q(^s), (33)

where is then the integrated cross section for the subprocess with both fractional quark/antiquark charges set to unity.

Above formula (32) is not very efficient when calculating subprocess energy distribution. A more useful formula is:

 σpp→~π0jj=∫dW(^σeffqq→q~π0q(W)∫dxd(Jqeff(x1,μ2F)qeff(x2,μ2F))). (34)

Above , , and is a Jacobian of the transformation from (, ) to (, ). In practical calculation first partonic (assuming elementary charges of quarks/antiquarks) is calculated as a function of on a grid and then the convolution with parton distributions is done as shown in Eq. (34). In the hadronic calculations we take .

## Iv Leading order VTC technipion signal in the diphoton channel

In the case of VTC technipion model Pasechnik:2013bxa (), the amplitude for the subprocess reads:

 Mγγ→~π0→γγ(λ1,λ2,λ3,λ4)=(ε(γ)μ3(p3,λ3))∗(ε(γ)μ4(p4,λ4))∗ ×ϵμ3μ4ν3ν4pν33pν44Fγγi^s−m2~π0+im~π0Γtot ×ϵμ1μ2ν1ν2pν11pν22Fγγε(γ)μ1(p1,λ1)ε(γ)μ2(p2,λ2). (35)

The can be calculated from a model or taken from recent experimental data. In the following we take the calculated value of and = 750 GeV. The mass scale of the degenerate techniquarks is in principle another free parameter (see e.g. Ref. Lebiedowicz:2013fta ()).

The cross section for the signal (see the left panel of Fig. 5) is calculated as ():

 dσdy3dy4d2pt,γ=116π2^s2∑ijx1γ(i)(x1,μ2F)x2γ(j)(x2,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mγγ→~π0→γγ|2, (36)

where

 x1=pt,γ√s[exp(y3)+exp(y4)],x2=pt,γ√s[exp(−y3)+exp(−y4)]. (37)

## V One-family walking technicolor model

In the one-family walking technipion model discussed recently in Ref. Matsuzaki:2015che () (see also references therein) the partial and decay widths are given as:

 Γ(P0→gg) = N2TCα2sGFm3P012√2π3, (38) Γ(P0→γγ) = N2TCα2emGFm3P054√2π3, (39)

where is the strong coupling constant, is the number of technicolors in the walking technicolor model. For = 3 we get:

1.2 GeV,    1.2 MeV.

The decay into two gluons is in this model the dominant decay channel Matsuzaki:2015che (). The total decay width in the model is therefore also much smaller than the 45 GeV reported in Refs ATLAS:2015 (); CMS:2015dxe () and used in many very recent analyses. We will return to this point in the result section. It is interesting that the model gives roughly correct size of the signal for = 3, 4, without any additional tuning.

The cross section for the signal in this scenario is calculated then as ():

 dσdy3dy4d2pt,γ=116π2^s21N2c−1x1g(x1,μ2F)x2g(x2,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mgg→~π0→γγ|2. (40)

The color factor guarantees that the technipion resonance is a QCD-white object. Clearly, in this model the gluon-gluon fusion is the dominant reaction mechanisms. This would also mean a rather large cross section for the dijet production of the order of a few pb. We shall show below whether this is compatible with the existing data for dijets production.

## Vi Production mechanisms of background in γγ channel

In the present exploratory analysis we consider the background contributions shown in Fig. 6. These include the annihilation (diagram (a)), the gluon-gluon fusion via quark boxes (diagram (b)), and the photon-photon fusion via lepton, quark and -boson loops (diagram (c)).

They were found recently to be very important for production with large Luszczak:2014mta (). In addition, this type of background could interfere with the signal. For simplicity we shall neglect these interference effects in the present paper.

### vi.1 Background q¯q annihilation contribution

The lowest order process for diphoton production is quark-antiquark annihilation. The cross section for annihilation can be written as:

 dσdy3dy4d2pt,γ=116π2^s2∑fx1qf(x1,μ2F)x2¯qf(x2,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mq¯q→γγ|2. (41)

The formula for the matrix element squared for the subprocess can be found e.g. in Ref. Berger:1983yi (). In our calculation we include only three quark flavours (, , ).

### vi.2 Background gg fusion contribution

For a test and for a comparison we also consider the gluon-gluon contribution to the inclusive cross section. The photons produced in are expected to be dominantly produced by the quark-antiquark annihilation () and by the gluon-gluon fusion () through a quark-box diagram. The latter process is important especially at low diphoton invariant masses in kinematic region with high gluon luminosity.

In the lowest order of pQCD the formula for inclusive cross section can be written as

 dσdy3dy4d2pt,γ=116π2^s2x1g(x1,μ2F)x2g(x2,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mgg→γγ|2. (42)

The corresponding matrix elements have been discussed in detail e.g. in Ref. Glover:1988fe ().

### vi.3 Background γγ fusion contribution

The cross section of production via fusion in collisions can be calculated in the same way as in the parton model in the so-called equivalent photon approximation as

 dσdy3dy4d2pt,γ=116π2^s2∑ijx1γ(i)(x1,μ2F)x2γ(j)(x2,μ2F)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mγγ→γγ|2. (43)

In practical calculations for elastic fluxes we shall use parametrization proposed in Ref. Drees:1994zx (). The loop-induced helicity matrix element for the subprocess was calculated by using the Mathematica package FormCalc Hahn:1998yk () and the LoopTools library based on vanOldenborgh:1989wn () to evaluate one-loop integrals. In numerical calculations we include box diagrams with leptons, quarks as well as with bosons. At high diphoton invariant masses the inclusion of diagrams with bosons in loops is crucial, see e.g. Lebiedowicz:2013fta ().

## Vii Discussion of results and consistency checks with existing experiments

### vii.1 Signal of technipion

Let us first summarize integrated cross sections for the VTC scenario. In Table I we have collected cross sections for different QED orders. For consistency all cross sections were calculated with the MRST(QED) parton distributions Martin:2004dh (). In this calculation we have used for example and our benchmark parameters. Surprisingly, different contributions are of the same order of magnitude. Note that with we get the cross section of correct order of magnitude. To describe the experimental signal more precisely can be rescalled. A result consistent with the cross section extracted from experimental ATLAS and/or CMS data is obtained with = 20.

We shall discuss now some specific results for different processes.

Let us start from the signal in the VTC model Pasechnik:2013bxa (). In our calculations here we use MRST04(QED) parton distributions Martin:2004dh (). In this () calculation the . For comparison . This means that for the purely exclusive reactions we get  fb. In order to get experimental value in the fiducial volume  fb we have to assume a rather large value of . This is a huge value and puts into doubts perturbative approach. We will not worry here about the conceptual problem and test further consequences. The corresponding  GeV (very narrow width scenario). The narrow width approximation was preferred by the CMS analysis CMS:2015dxe ().

As discussed in section III.3 the production of technipion can be calculated also as subprocess with intermediate (off-shell) photons. We neglect here intermediate exchanges for simplicity. We shall discuss now how such results compare to the previous results obtained in the () or () when photons from the decay of neutral technipion are considered, and “initial” photons are assumed to be on-shell.

In Fig. 9 we show the rapidity distribution. We show here both the contribution of (see Fig. 1) and two contributions of (see Fig. 2) processes. Each of the ( and ) contributions separately is asymmetric with respect to = 0. The sum is then similar as for the contribution. The calculation of rapidity distribution of for the mechanism with subprocess (see Fig. 3) is more complicated and will be omitted here.