Multi-Scalar Production At LargeCenter-Of-Mass Energy

Ali Shayegan Shirazi

Department of Physics,Sharif University of Technology,

Tehran 11155-9161, Iran

Abstract

In quantum field theory, the probability of producing scalar parti-cles grows factorially as a function of the number of the particlesproduced. This poses a problem theoretically, in maintaining uni-tarity, and is counter-intuitive phenomenologically. The factorialgrowth is a byproduct of the perturbation theory. Nevertheless,it has been recently proposed that the factorial growth might beobservable in the future 100 TeV hadron collider. We collect someof the calculations that had been done in regards to this problemso far. We then find the ratio σn/σtotal by calculating the numberof scalar jets one would observe at high center-of-mass energies.We will present our results for φ3 theory in four and six space-time dimensions, φ4 and the broken theories in four spacetimedimensions.

Contents

1 Introduction 1

2 Review of the Past Calculations 32.1 Generating Function Method . . . . . . . . . . . . . . . . . . . 32.2 Exponentiation and The Holly Grail Function . . . . . . . . . 52.3 Semi-Classical and Non-Perturbative Methods . . . . . . . . . 62.4 Enhancement of Multi-Higgs Cross-Section via Higgsplosion . 8

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3 Jet Generating Functional Method 103.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 φ3 Theory In Six Spacetime Dimensions . . . . . . . . . . . . 11

3.2.1 IR Divergences, AP Function, and The Sudakov FormFactor . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.2 Jet Generating Function . . . . . . . . . . . . . . . . . 153.3 φ4 Theory In Four Spacetime Dimensions . . . . . . . . . . . . 16

3.3.1 Cancelation of IR Divergences . . . . . . . . . . . . . . 163.3.2 AP Function and The Sudakov Form Factor . . . . . . 203.3.3 Jet Generating Function . . . . . . . . . . . . . . . . . 21

3.4 φ3 Theory In Four Spacetime Dimensions . . . . . . . . . . . . 223.4.1 Cancelation of IR Divergences . . . . . . . . . . . . . . 223.4.2 AP Function and The Sudakov Form Factor . . . . . . 273.4.3 Jet Generating Function . . . . . . . . . . . . . . . . . 29

3.5 The Broken Theory In Four Spacetime Dimension . . . . . . . 29

4 Off-sehll φ∗ → nφ Process. 31

5 Comparison to Fix-order Calculation 33

6 Conclusion and Discussion 36

A Loop Corrections to All Order at Threshold for φ3 Theory inSix Spacetime Dimensions 38

B AP Function for φ3 Theory in Four Spacetime Dimensions 41

C AP Function for φ4 Theory in Four Spacetime Dimensions 43

2

1 ◦ Introduction

It has been known for a while that the production of many-particle states athigh energies can become a factorial function of the final number of particles.This problem drew attention in the Electro-Weak theory when it was foundthat the B+L-violating cross-section, via instanton-like processes, becomelarge at high energies with large number of bosons in the final state [1, 2].Motivated by this discovery, the effect was analyzed in simpler models i.e.scalar theories to find whether the ever-increasing cross-section is actuallyphysical and what is the upper bound on the cross-section without violat-ing unitarity [3, 4]. Multiple methods were used in order to find either theamplitude or the cross-section with many bosons in the final state. Usinggenerating function method, Brown found the exact amplitude at thresholdfor any number of final states [5]. The method were applied to the loops andsumming the loop corrections to all order [6–9]. Other methods includes,but are not limited to, Recursion relations [10, 11] Coherent state formal-ism [12, 13], instanton calculation (Lipatov method) [14–16], and FunctionalShrodinger equation [17, 18].

Most of these results are from before LHC, where we did not observe anyB-L violation. In the past few years, however, perhaps because the 100 TeVcollider is getting closer to reality, the multi-particle cross-section, in partic-ular multi-Higgs cross-section, has received some new attention [13, 20–24].According to [23], the factorial growth is in fact physical and observable inexperiment. Consequently, they foreseen new phenomenon in high energycolliders (of order 100 TeV or so) with many-particles signature. This phe-nomenon has been dubbed “Higgsplosion”. They propose that the unitaritycan be preserved through a mechanism called “ Higgspersion”. Moreover, itis claimed that these mechanisms can solve the Hierarchy problem.

Whether the Higgsplosion is physical and the calculation is conclusive hadbeen subject of debate [25–28]: The inclusion of the heavy fermion loops canchange the amplitude at high multiplicity of particles, unitarity might notbe actually preserved in this mechanism, etc. We also believe that there is agreat weight given to the perturbative calculation which in our opinion shouldnot be trusted at the point where the amplitude becomes large (thought, it isclaimed that non-perturbative calculation valid for any λn also gives factorialgrowth).

Most of the calculations referred above are based on the multi-scalarcross-section near threshold, i.e. when the final particles are non-relativistic

1

and are at fixed angle with respect to each other, hence free of collineardivergences. This will confine these results to a small corner of the phase-space. What we will try to do in this paper is to focuses on that region ofphase-space where there are enhancements in the amplitude due to collineardivergence’s. That is, we want to address the phenomenology of multi-scalarby finding the cross-section for producing n scalar jets. This is done using JetGenerating Functional (JGF) [29, 30], which satisfies an equation analogousto Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [31, 32].These calculations are semi-non-perturbative in nature, and as a result weexpect them to be accurate in the large multiplicity limit.

The main object in our calculation is the Sudakov form factor (SF) [33,34], which is the probability that a particle would not split into two or moreparticles, i.e. the probability that it stays distinct, given the resolution ofthe experiment. We start with a particle at energy of order t ≈ Q2, whereQ is the hard scale, e.g. the center-of-mass energy, and an IR regulator t0,which can be the minimum opening angle required to resolve two lines, i.e.t0 = tδ2, or, as we will discuss further below, in the massive theory, if theopening angle is not too large, it is t0 = m2. Given these two scales, theSudakov factor, ∆, is of the form

∆(t, t0) = exp

(−∫ t

t0

dzP(z, t′)dt′). (1)

where P is the Altarelli-Parisi (AP) splitting function and z is the energyfraction carried by the daughter particle.

We will find the AP function and SF for the cubic theory in four and sixspacetime dimensions; and for the quartic theory, broken and un-broken, infour dimensions. We will then use these to write a JGF for these theoriesand use it to find the jet rates.

Given that the previous calculation is relevant to fixed angle betweenparticles and for particles carrying small amount of energies, our result isnot in one to one correspondence with the perturbative calculation that leadsto the factorial growth. However, our result does not show any sign of thefactorial growth in the relativistic limit, while if we believe that the factorialgrowth should become physical in one point of phase-space, there is no reasonto believe that it would not happen in other regions.

In section 2, we briefly review the perturbative and semi-classical/non-perturbative methods that had been used in the past in regards to multi-scalar cross-sections. We also review the phenomenology of the Higgsplosion,

2

where it is claimed that the multi-Higgs production become unsuppressed dueto the factorial growth and observable in the future colliders. In section 3,we discuss the jet generating function method and apply it to the scalartheories in six and four spacetime dimensions. In section 4 we plot themost likely number of final particles in the quartic and cubic theories in theprocess off-shell φ∗ → nφ, and in section 5 we compare to fix-order numericalcalculation.

2 ◦ Review of the Past Calculations

2.1 Generating Function Method

Brown [5] has utilized an elegant and simple method for finding the amplitudefor an off-shell scalar to produce n on-shell particles, when the particles areat threshold. The idea is that when the final particles are at threshold, thegenerating function of the tree amplitudes is the classical solutions of theequation of motion in the presence of a source term, and using the classicalsolution one finds the amplitude as the coefficient of the series in the source.

In the unbroken φ4 theory,

L =1

2(∂φ)2 − 1

2m2φ2 − 1

4!λφ4, (2)

the classical solution is,

φcl =z

1− (λ/48m2)z2, unbroken sym. (3)

where z is proportional to the source. The amplitudes are found by differen-tiating with respect to z and setting z = 0. We find,

Atreen = 〈n|φcl|0〉 = n!( λ

48m2

)n−12, unbroken sym. (4)

with n = 3, 5, . . . . Consequently, the cross-section, once multiplied by1/n! for accounting for the n identical bosons in the final state, will growfactorially.

3

In the broken theory1, one finds [6]

φcl =z

1− z√λ/12m2

, broken sym. . (6)

Here the amplitude will be

Atreen = n!( λ

24m2

)n−12, broken sym. . (7)

Again, with a factorial growth. Loop correction does not alleviate the prob-lem. In [6, 7], using Brown’s method, and in [10] by using recursion relationbetween the one loop diagrams, thy had been calculated 2. In both casesone needs to choose the counter-terms appropriately, otherwise no analyticsolution can be found. For the unbroken theory, the leading n dependence ofgenerating function has been found to the first order in perturbation theoryto be

φcl =z

1− (λ/48m2)z2

(1− λ

4B

(λ/48m2)2z4

(1− (λ/48m2)z2)2

), unbroken sym.

(8)where B is given by

B =

√3

2π2

(ln

2 +√

3

2−√

3− iπ

). (9)

The amplitude is modified to

An = Atreen

(1− λB

32(3− 2n+ n2)

)unbroken sym. (10)

The fact that the loop correction vanishes for n = 3 is the result of thesubtraction scheme used.

For the broken case the results are as follow

φcl =z

1− z√λ/12m2

(1− λ3/2z

48πm(1− z√λ/12m2)2

)broken sym.

(11)1The solution is not unique, one also finds that

φcl = φ01 + z/2φ01− z/2φ0

, (5)

with φ0 =√

3!m2/λ works too [5].2For a review, see appendix A where we calculate the loop correction for the cubic

theory in six spacetime dimensions.

4

2.2 Exponentiation and The Holly Grail Function

Later on, from analyzing the singularities of the generating function for higherorder corrections, it was shown that these loop corrections exponentiate forλn < 1 [8]. If we write An = Atree

n A1n with Atree

n given in (4) and (7), we have

A1n = exp

[− λn2 1

64π2

(ln(7 + 4

√3)− iπ

)]unbroken sym.

A1n = exp

[λn2

√3

8π

]broken sym.

.These results had led to the conclusion that the cross-section can be

written as the exponential of a function of λn, and dubbed the Holy Grailfunction:

σn ∝ enF (λn) (12)

where the Holy Grail function, to first order, is

F = logλn

48− 1− λn2Re[B]

32π2unbroken sym. (13)

F = logλn

24− 1 + λn

√3

4πbroken sym. (14)

The higher loop corrections will be of order (λn)2 or greater. We cansee that the tree calculations are a good approximation as long as λn < 1.For λn ≈ 1, the perturbation series will blow up and we need to considernon-perturbative approaches which we shall discuss in the next sections.

The threshold limit corresponds to the limit where the kinetic energy ofthe final particles vanishes. Unlike the amplitudes founds so far, the cross-sections at the threshold actually is zero, since there are no phase-spaceavailable. To find the cross-section slightly away from threshold, where wecan still use the amplitudes, we will work in the approximation where all thefinal particles have the same average kinetic energy

ε =E − nmnm

(15)

where E is the energy of the incoming off-shell particle.

5

Since the amplitudes are space-time independent, the cross-section is ap-proximately give by

σn ≈1

n!|An|2

(ε3/23π

)n(16)

As for the amplitudes, the energy dependence is found by recursion relations.For small energies one finds that [9]

An = Atreen e−56nε. (17)

Integrating the square of the amplitude over the phase-space near threshold,the total contribution to the Holy Grail functions will be

F =F0 + f(ε) (18)

f(ε) =3

2(log ε/3π + 1)− 17

12ε (19)

Where F0 is given by (13) and (14). This expression is valid for λn < 1 andε� 1.

2.3 Semi-Classical and Non-Perturbative Methods

There has been considerable number of attempts to address the multi-scalarproblem in 90s using different methods. For completeness, let us brieflydiscuss some of them and refer to the original works for further details.

• The Coherent State Formalizm approach is based on the steepest decentmethod using coherent states in QFT and is similar to Laundau WKBmethod in quantum mechanics. The application to multi-particle hadbeen pioneered by Son [12] and more recently studied in more detail in[13]. The result for the loop corrections in the λn� 1 is

F = log λn− 1 + 0.85√λn broken sym. (20)

As far as the energy dependence is concerned, in the ε � 1 limit, thenext to leading correction is [9]

f(ε) =3

2

(log ε/3π + 1

)− 17

12ε+

1327− 96π2

432ε2 (21)

6

Furthermore, using this method in ε → ∞ limit, Son [12] had shownthat the cross-section satisfy the lower bound3

σn > n!( λ

48π2

)n(22)

• The Lipatov Method approach is based on analytically continuing tothe negative values of λ = −λ′, where the potential will be invertedand all the amplitudes will acquire an imaginary part. One can writethe real part of the amplitude for positive λ in terms of the imaginarypart for the negative λ, through the dispersion relation

An(pi, λ) = const.+λ

π

∫ ∞0

dλ′Im[An(pi, λ

′)]

(λ′ + λ)λ′(23)

For negative λ, the imaginary part of the amplitudes can be calculatedusing the instanton solution of the inverted potential [14]. The authorsof [15, 16] have applied this idea to the amplitude An(2 → n − 2), inthe scalar theory. Writing the amplitude as

An =∑l

alnλn/2−1+l (24)

where l is the loop order (l = 0 is the tree level). The coefficients alncan be calculated by expanding (23):

aln = (−1)n/2+l1

π

∫dλ′

Im[An(pi, λ′)]

λ′n/2+l(25)

It is shown that in this method the expansion parameter is4

η =n

n+ l. (26)

And that the energy dependence of the amplitudes becomes importantas η → 1. Hence, since the expansion parameter should remain small,

3Although this result is based on semi-classical calculation, it is in contradiction ofwhat we claim to be the case.

4Given that the integral (25) peaks at λ′ = 16π2

n+l , the expansion parameter is η ∝ λ′n,which is reminiscent of the expansion parameter found earlier.

7

the results in this method should be trusted when the final particlesare near threshold.

It is shown that the factorial behavior shows up in this method as well:

aln ∝(l + n/2)! l� 1, n = O(1) (27)

aln ∝Γ(l + n+ 3/2) l� n� 1 (28)

• Yet another attempt had been done using Functional Schrodinger Method.The most important outcome is that the cross-section should not growat high multiplicity of the final particles. For example, it can be shownthat the amplitude for nλ � 1, the cross-section decays exponen-tially5 [17, 18]

σn ∝ exp(−π2n). (29)

2.4 Enhancement of Multi-Higgs Cross-Section via Hig-gsplosion

Finally, let us review the most recent work on the observation of multi-Higgscross-section [20–24]. For observing the factorial growth in the experiment,one needs to find the energy dependence away from threshold limit. Non ofthe method known gives a good approximation in large ε limit, that is whenthe final particles are relativistic. It is, however, possible to extract theepsilon dependence at tree level using Monte Carlo simulation if we acceptthe ansatz that the cross-section will exponentiate into a holy grail form (12),specifically that the dependence on λ comes in λn form[21].

The point is that, at tree level, the f(ε) does not depend on λ, and sincethe expansion parameter is λn, it will not depend on n either. Hence, if welook at the ratio of two consecutive cross-sections, we find that

log σn+1/σn = (n+ 1)F0(n+ 1)− nF0(n) + f(ε) (30)

Since F0 is known, the authors of [21] used Madgraph [49] to find a fit for f(ε)for n = 5. We have not redo their simulation; instead used a fit to their graphin figure 2 of [21] to display their result here to complete this section. Weused the fit together with the expression in (20) for the broken theory (notethat we do not have a non-perturbative equation for the unbroken theory), to

5This can also be shown in a quantum mechanical system [19].

8

produce Fig. 1 (compare to figure 6 in [20]; however, in this paper the authorsused the perturbation result for the loop correction (14) which is not valid atlarge λn). We can see that the cross-section becomes un-suppressed at finitevalues of ε before they fall down due to suppression from the phase-space.

��� ��� ��� ����

-����

-����

-����

-���

���

����

����

��� σ�

Figure 1: The multi-Higgs phenomena. From top to bottom, the center-of-mass energy is 100, 50, 10 TeV. The suppression at the far right is due to thephase-space, where particles are non-relativistic.

It is important to note that the enhancement of the cross-section atO(100) TeV energies and low number of final particles (what is suggestedto look for in Multi-Higgs papers) is not so much a result of the factorialgrowth but the fact that the correction to the tree result had come with apositive sign in (20). If we naively just use the tree result, we would have anenhancement at much higher energies ≈ O(104) TeV.

Furthermore, the cross-section becomes un-suppressed at values of ε forwhich the final particles are relativistic. As explained earlier, we will use jetgenerating function method to sum the contribution for producing n jets.

9

3 ◦ Jet Generating Functional Method

3.1 Overview

Jet generating functional (GF) [29, 30] is a functional of a probing function,u(pi), where pi is the momentum of i’th particle, such that when expandingin u, the coefficients are the exclusive particle cross-sections. Here, we willuse the simplified version where we had integrated over the momentums ofthe particles to obtain jets. So that the coefficients are jet rate and we havea function instead of a functional.

Calling the generating function Φ, by definition the jet rates are

Rn =σn−jetsσtotal

=1

n!

( ∂∂u

)nΦ[u, t]

∣∣∣u=0

. (31)

where t is the energy of the incoming particle that produces jets.One can also use this function to find the average multiplicity number as

follow

n =∑

nRn =∂

∂uΦ[u, t]

∣∣∣u=1

. (32)

In our understanding, there is no real derivation of the generating func-tional from first principles. We had found that Chang and Lau [35], and laterTaylor [36], were probably the first ones to apply the idea to φ3 theory insix spacetime dimensions. Later, Kinoshi and others [37] used Altarelli-Parisievolution equation [31, 32, 39, 40] to “derive” an equation for gluon and quarkjet GF, mainly based on jet evolution from a perturbation understanding.

For now, let us work with a cubic theory, whether QED or QCD, or φ3

theory. The main equation for a JGF can be written as follow

Φ[u, t] =Φ[u, t0]∆(t, t0) (33)

+

∫ t

t0

dt′dz∂P(z, t′)

∂z∂t′∆(t, t0)

∆(t′, t0)Φ[u, z2t′]Φ[u, (1− z)2t′]. (34)

where the z is the energy fraction of one of the daughter particles. Thisequation gives how a line at scale t evolves and splits into other lines. A jetwith a large energy wants to split into more jets unless its energy is of ordert0, where by definition it cannot split anymore: Φ(u, t0) = u. The first termon the right hand side describes the line if it had not split into other jets.

10

The ∆(t, t0), hence, is the probability of not splitting given the two scales, t0,and t. It is called the Sudakov form factor [33, 34] . The second term on theright hand side is the sum (turned into an integral) of probabilities of the linesplits into two other jets at scale t′. The P function is the probability weightof splitting into two lines with energy fractions z and 1 − z. The fraction∆(t, t0)/∆(t′, t0) ≈ ∆(t, t′) is the probability that the line had not splattedbefore splitting at scale t′.

We had not yet defined the variable t. It is claimed, and usually taken tobe,

t =k2T

z2(1− z)2, (35)

where KT is the transverse momentum of daughter jets.. For a two-bodydecay, it is

t = Q2(1− cos2 θ), (36)

where the angle is between the decayed particles. It is claimed that thegenerating function that satisfies (33), with the so defined t, correctly sumsthe divergent logs for jets in KT -algorithms [38].

In this paper, we will use t = K2T . Given the limits we take and approx-

imation we make, there will be not much difference, and we will make notewhere there will be. Furthermore, we will argue that in the massive theoryin t0 → 0 limit, where t0 is the resolution of the experiment, in the finalexpression we can change t0 to mass and t to Q2, the hard scale energy (Ashad been done in the earlier versions of the generating functional [37]).

In QCD, (33) gives the jet rate which phenomenologically gives differentscaling for abelian and non-abelian splittings: Poisson pattern and staircasepattern respectively, that can be examined in experiment [41]. Our maingoals would be to find P and ∆ in the scalar theory, and plug them into (33)and use (31) to find the jet rates.

3.2 φ3 Theory In Six Spacetime Dimensions

The cubic scalar theory in six dimensions provide a good working ground foranalyzing the multi-scalar problem. This is due to two facts. First, unlike infour dimensions, the coupling is dimensionless6. And, secondly, each particle

6The theory is asymptotically free, which is why it used to be a toy model for gluons.

11

can split into two particle, which makes finding the splitting functions andSudakov factor easier than in quartic theory where each particle can onlysplit into 3 particles in perturbation theory.

The Lagrangian is given by

L =1

2(∂φ)2 −m2φ2 − g

3!φ3 +

1

2δZ(∂φ)2 − 1

2δmφ

2 − 1

3!δgφ

3 − δτφ. (37)

The multi-particle amplitude via the classical generating function methodused by Brown is [11]

φcl =z

(1− λ12z)2

φ3in 6D (38)

This gives the amplitude at tree level,

A1→n = nn!

(λ

12m2

)n−1

, φ3in 6D (39)

where we have restored the mass. Again, we see a factorial growth whichis expected since the tree graphs amplitude basically counts the number ofthe Feynman diagrams.

We calculated the loop correction in appendix A. The result is

An =dn

dxn(φ0 + φ1)

∣∣x=0

= nn!( g

12

)n−1[1 + 3g2(n2 + 3n+ 2)

]φ3in 6D

(40)

Interestingly, we find that the conjecture that the expansion is in gn (here(gn)2), holds here as well.

3.2.1 IR Divergences, AP Function, and The Sudakov Form Fac-tor

Under Bloch-Nordsieck theorem [42] and Kinoshita-Lee-Nauenberg theorem[43], in QED and QCD respectively, the mass singularities cancel in the totalcross-section between the virtual contribution and the radiation contribution[44]. This phenomenon happens for the scalar theories too. Here we willfollow Srednicki [45] who had shown the cancelation for the scalar cubictheory in six dimensions and we will show it for the cubic and quartic theoriesin four dimensions below.

12

Following Srednicki, for the exclusive process φφ → φφ, in the m → 0limit, we need to use MS scheme instead of un-shell scheme. In this schemeand in the mass-less limit, the 1PI loop corrections do not depend on massafter renormalization:

M = R2M0

[1− 11

12

g2

(4π)3

(log

s

µ2+O(m)

)+ . . .

](41)

In this scheme, however, the residue of the two-point function at the masspole is not one. The residue, R, comes from LSZ formula, with each legcontributing

√R. We have

R =1

1− Π′(mphy)≈ 1

1− Π′(m)≈ 1− 1

12

g2

(4π)3log

µ2

m2(42)

where prime is differentiation with respect to p2. Altogether, we find,

M =M0

[1− g2

(4π)3

(11

12log

s

µ2+

1

6log

µ2

m2+O(m)

)+ . . .

](43)

These logs would not sum by renormalization of the coupling constant; wecannot remove them by changing µ. They cancel when we add to the cross-section the radiations off of the legs. The radiation contribution to the cross-section is

1

2!

∫g2

(p2a −m2)2d6pbd

6pc (44)

where the line a has splatted to b and c as shown in Fig. 2. A delta function isadded at this point to turn one of the phase-space integrals into the integralover pa, so that we can factor out the cross-section of one less particle process.Carrying out the integral under the assumptions m2 � Q2 and m2/Q2 < δ2,where δ is the resolution angle, we find

g2

2(4π)3

∫ 1

0

z(1− z)dz

∫ δ

0

θ3dθ

(θ2 + (m2/E2a)f(z))2

=g2

12(4π)3log

δ2E2a

m2+ . . .(45)

where f(z) = (1 − z + z2)/(z − z2)2. There is no soft divergence, just thecollinear divergence. Hence the single log. This is just what we need for

13

a

b

c

Figure 2: φ→ φφ. Momentums are flowing to the right.

canceling the log from the virtual diagrams and turning it into a log of δ.Using E2

a = s/4, we find

σ =(

1 +g2

12(4π)3log

δ2E2a

m2

)4|M|2 (46)

=M20

[1− g2

(4π)3

(3

2log

s

µ2+

1

3log

1

δ2O(m)

)+ . . .

](47)

The g2

12(4π)3log δ2 can be interpreted as the probability of a leg not split-

ting. The fact that it blows up instead of reaching 1 as δ → 0, is a sign of thefact that we need to re-sum the perturbation series. This is usual practicein QED and QCD, and can be shown for cubic scalar in six dimensions too,by summing further radiations of the leg, or purely on probabilistic grounds[44]. The result is the modulation of the cross-section with the Sudakov formfactor

∆ = exp[− g2

12(4π)3log[1/δ2]

](48)

for each external leg. The forth power of Sudakov form factor, when ex-panded, would give the log in (45).

The remainder of the integral in (45) is nothing but the probability ofa line splitting. Integrated over the range of θ where the opening angle isbigger than the resolution, we find

g2

2(4π)3

∫z(1− z)dz

∫ 1

δ

θ3dθ

(θ2 + (m2/E2a)f(x))2

≈ g2

12(4π)3log

1

δ2(49)

Let us note that these results are correct for m� Q, i.e. in the masslesslimit. What if m2

Q2 > δ2? In this limit, the integral in (45) vanishes as δ → 0,

14

and the radiation probability becomes ≈ g2

12(4π)3logQ2/m2. In the massless

limit, the Sudakov form factor sums the radiation contribution and virtualcorrections in such a way to give the probability of not splitting given somejet definition (loosely, we can change δ with other jet definition, for examplejet mass or trust). So we expect that for δ < m/Q, it becomes

∆ = exp[− g2

12(4π)3log

Q2

m2

]. φ3 in 6D (50)

This can actually be shown in Soft Collinear Effective Theory [46], thatin a massive theory the Sudakov form factor sums the IR divergent logs. Inother words, we can think of the mass as the IR regulator, since we find thesame expression by changing δ to m.

From now on, we will set m = 0 whenever possible, and use the symbol t0for the IR scale. If m/Q > δ, t0 = m2; if m/Q < δ, t0 = δ2Q2. We will alsouse the symbol t for Q2 interchangeably. We also note that in both limits,we still have m2/Q2 � 1, or t0/t�< 1

Using the definitions and approximation above, we can find the probabil-ity density of one line splitting into two lines now, called the Altarelli-Parisifunction:

∂P∂z∂t′

=g2

2(4π)3z(1− z)

t′(51)

Knowing the Alterali-Parisi, we can form the Sudakov form factor. It isthe exponential of the AP function:

∆(t, t0) = exp[−∫ 1

0

dz

∫ t

t0

dt′∂P∂z∂t′

](52)

3.2.2 Jet Generating Function

We now have all the ingredients for writing the differential equation for thegenerating function. For the cubic theory it is

Φ(3)[t] =u∆(t, t0)

+1

2(4π)3

∫ t

t0

dt′∆(t, t0)

∆(t′, t0)

∫dzg2z(1− z)

t′Φ(3)[z

2t′]Φ(3)[(1− z)2t′]

(53)

15

Noting that the distribution in z is fairly even, we disregard the z-dependence of the functions under integral to find the simplified equation,

Φ(3)[t] =u∆(t, t0) (54)

+1

2(4π)3

∫ t

t0

dt′∆(t, t0)

∆(t′, t0)

∫dzg2z(1− z)

t′Φ(3)[t

′]Φ(3)[t′]. (55)

Now, differentiation with respect to t from both sides gives

Φ(3)[t]

dt=

1

∆

d∆

dt

(Φ(3)[t]− Φ2

(3)[t]). (56)

Using the boundary condition, Φ[t0] = u, we find

Φ(3)[t] =u

u+ (1− u)∆−1(57)

This is in agreement with Taylor [36] and Kinoshi [37]. This also is theJGF for a gluon in pure Yang-Mills theory, that is without splitting of thequark and anti-quarks into gluons and vice versa, which leads to staircasepattern for the gluons [41] that had been checked in experiment.

Using (57) and (31), we find that

Rn = ∆(1−∆)n−1. (58)

Using (32), we can find the average jet multiplicity

n =∂

∂uΦ∣∣∣u=1

= ∆−1. (59)

As g → 0, we have n → 1, which means that there are no splittings; on theother hand n increases with Q2/m2, or decrease with δ, as expected.

3.3 φ4 Theory In Four Spacetime Dimensions

3.3.1 Cancelation of IR Divergences

We now turn to the φ4 theory

L =1

2

(∂φ)2 − 1

2m2φ2 − λ

4!φ4. (60)

16

We want to analyze the scattering process X → φ→ 3φ - where X can be, forexample, e+e−, and the intermediate φ is off-shell. The infrared divergencesshould cancel between the virtual diagram of the X → 3φ process againstthe divergences in X → 5φ process. The virtual diagrams are at two loopslevel. One might suspect that the “fish diagrams” would contribute as well.However, the fish diagram does not contain any IR divergence. This shouldbe the case since the contribution ofX → 5φ process to the total cross-sectionis at λ4 order, while, if the fish diagram has IR divergence, there would be aλ3 order contribution to the total cross-section (from the interference of thefish diagram with the tree diagram).

Checking the IR cancelation in the total cross-section for the process X →3φ requires calculating 5-body phase-space. We instead check the cancelationof the mass-divergence of the 2 loop diagram in 2φ→ 2φ scattering (Fig. 3)via the Srednicki method.

Figure 3: φφ→ φφ.

We want to find the 1PI four-point function in the m2 → 0 limit. Thevirtual diagrams are shown in Fig. 4.

(1) (2) (3)

Figure 4: Types of virtual corrections to φφ→ φφ.

As note in [45], the correct renormalization scheme to use in zero masslimit is the MS scheme. In this scheme, the residue at the physical mass poleis not unity and we have a factor of

√R multiplying the cross-section for

each external leg coming from LSZ formula. Where

R =1

1− Π′(p2 = m2). (61)

17

In this scheme, the leg loop correction, i.e. the sunset diagram (Fig. 5), isnot absorbed in the counter term and needs to be added to the cross-section.

Figure 5: The “sunset” diagram.

The “fish” diagrams with the counter term added, in MS scheme, in them→ 0 limit, gives

M(1) =−iλ2

2(4π)2

∑A=s,t,u

(log

A

µ2− 2)

(62)

where µ is the renormalization scale and s, t, and u are the Peskin parameters.The double “fish” diagrams, in the same limit, gives

M(2) =−iλ2

4(4π)4

∑A=,s,t,u

(log

A

µ2− 2)2

(63)

The “vertex correction” diagrams give

M(3) =−iλ3

(4π)4

∑A=s,t,u

(1

2log2 A

µ2− 3 log

A

µ2+

11

2

)(64)

The “sunset” diagram gives

R =1

1− Π′(m2)≈ 1 + Π′(m2) = 1 +

λ2

12(4π)4log

m2

µ2+ const.×O(λ2)(65)

The cross-section is proportional to

R2|M |2 = λ2R2∣∣∣1 +M(1) +M(2) +M(3)

∣∣∣2. (66)

We note that in MS scheme, all that survive in m→ 0 limits are the logof m in R. The amplitude depends on the external momentum and µ (the

18

cross-section does not depend on µ by the running of λ) but not on the mass.This is interesting since we can now see that that IR divergences should beat λ2 order since there is no field renormalization in the quartic theory atone loop level. Furthermore, since R only depends on logm, we expect thatthere are no double logs at this order for the quartic theory either.

To find the physical amplitude in the m → 0 limit, we need to add to(66) the probability of the legs not split given resolution angle δ.

We start with a line, labeled a, splitting into 3 other lines, b, c, and d, asshown in Fig. 6.

a c

d

b

Figure 6: φ→ 3φ splitting. Momentums are flowing to the right.

The splitting is proportional to

1

3!

∫λ2

(p2a −m2)2d4pbd

4pcd4pd (67)

we multiply this by

1 =

∫d4pa2Ea(2π)3δ(~pa − ~pb − ~pc − ~pd), (68)

and isolate the d4pa integral to be absorbed into the cross-section withtwo less legs. We find

λ2

48(2π)6

∫Ea

EbEcEd(p2a −m2)2sin θbdφb sin θcdφcE

2bE

2cdEbdEc (69)

We can simplify this integral by changing the variables to: θ, α, x, and y.Where θ is the angle between b and c; α is the angle between d and the sumof b and c; and x and y are defined as follow

Eb = (1− x)(1− y)Ea (70)

Ec = x(1− y)Ea (71)

Ed = yEa (72)

19

The integral becomes

λ2

48(2π)4

∫ 1

0

dx

∫ 1

0

dy

∫ δ

0

θdθ

∫ δ

0

αdα−x(1− x)(1− y)

y×(

θ2 +x(1− x)

yα2 +

1

2yx(1− x)

m2

E2a

)−2(73)

We first integrate over α and θ using mathematica. We then expand inm and carry out the rest of the integrals to find

λ2

24(4π)4log

E2aδ

2

m2. (74)

Adding four of these to (66) exactly cancels the mass divergence thatcomes from R2:

R2 + 4× λ2

24(4π)4log

E2aδ

2

m2= 1 +

λ2

6(4π)4log

δ2E2a

µ2(75)

3.3.2 AP Function and The Sudakov Form Factor

It is not as straight forward to find an expression for the AP function asit was in the cubic theory in six dimension (or as it is in QED and QCD).It is mainly because we cannot separate the integrals over the angles fromthose of the energies. The AP function is usually written in terms of energyfractions of the daughter partons and their transverse momentums. Usingthese, in appendix C, we have found that the AP function, in the masslesstheory, is

P(x, z, t, t′) =λ2

16(2π)4xz(1− x− z)[x(1− x)t+ z(1− z)t′][(x(1− x)t+ z(1− z)t′)2 − 4x2z2tt′

]3/2 (76)

where x = Eb/Ea and z = Ec/Ea. The integration with these choice ofvariables is not easy. But, from our other choice of variables in (73), we knowthat it gives ∫

P(x, z, t, t′)dzdxdtdt′ =λ2

24(4π)4log

E2aδ

2

m2. (77)

20

Knowing the AP function, we find the Sudakov factor:

∆(t, t0) = exp[−∫P(x, z, t, t′)dzdxdtdt′

](78)

= exp[− λ2

24(4π)4log

t

t0

]φ4 in 4D (79)

where t is the large scale where a hard process starts.

3.3.3 Jet Generating Function

We are now ready to write down the JGF for the quartic theory. In this casewe need to take into account that a line splits into at least 3 other lines. Itreads

Φ(4)[t] =∆[t, t0]Φ(4)[t0] (80)

+

∫dx

∫dz

∫ t

t0

dt′∆(t, t′)P(x, z, t′)× (81)

Φ(4)[x2t′]Φ(4)[z

2t′]Φ(4)[(1− x− z)2t′] (82)

Since, like the cubic theory, there is no IR divergence in the energies, andso the integrand is not concentrated around x, z ≈ 0, 1, we again make thesimplification of ignoring the energy dependence of the functions under theintegral. Using ∆(t, t′) = ∆(t, t0)/∆(t′, t0), and differentiating with respectto t from both sides, we find

dΦ(4)[t]

dt=d∆(t, t0)

dt

(Φ(4)[t]− Φ3

(4)[t])

(83)

Solving this equation with the boundary condition Φ[t0] = u, we find

Φ(4)[t] =u√

u2 + (1− u2)∆−2(t, t0)(84)

Using mathematica, we find that

Rn = f(n)∆(

1−∆2)n−1

2n = odd ≥ 3, (85)

with f(n) a slowly decreasing function of n as shown in Fig. 7.

21

50 100 150 200n

0.1

0.2

0.3

0.4

0.5

f(n)

Figure 7: f(n) in equation (85).

3.4 φ3 Theory In Four Spacetime Dimensions

We have presented the perturbation result for the broken theory in section 2.Before we discuss the jet rates in this theory, let us briefly look at the cubictheory in four dimensions. The Lagrangian is

L =1

2

(∂φ)2 − 1

2m2φ2 − g

3!φ3 (86)

We can assume there exists a λφ4 term with a negligible quartic coupling λ�0 to avoid an unstable vacuum. Since in a general scalar theory, the cubicand the quartic couplings do not necessarily correlate, the IR divergencesfor radiations through cubic coupling should cancel independent of quarticcoupling.

3.4.1 Cancelation of IR Divergences

To show the cancelation, for convenience, we set m = 0 and use dimensionalregularization to regulate both the UV and the IR divergences [48]. Let usfind the total cross-section of the process e+e− → φ→ X, where φ is off-shell:

σtotal =R2σb + σv + σr. (87)

Here R is the field renormalization, b stands for Born, v for virtual, andr for the real emission contributions. In the limit m → 0, keeping g fixed,IR divergences should cancel as shown in Fig. 8.

22

×∗

←→ ×∗

×∗

←→

∣∣∣∣∣∣∣∣∣∣2

Figure 8: Cancelation of IR divergences between virtual (left) and real (right)contributions to the total cross-section.

Since the final particles are identical, we need to divide the cross-sectionsby n!, where n is the number of the final particles. Calling all the pre-factorsthat depend on the production of φ, Ab (in the example above, e+e− → φ),we have

σb = Ab1

2!

∫Π2|M0|2 (88)

σv = Ab1

2!

∫Π2|δMv|2 (89)

σr = Ab1

3!

∫Π3|MR|2 (90)

The divergent part of the field strength tensor, R, is proportional to∫d4k

k4=

1

ε− 1

ε′. (91)

The ε is due to IR divergence and ε′ due to UV divergence. Later on, bothwill cancel separately in the total cross-section. Bu, for convenience, insteadof keeping both UV and IR regulators, we can instead set ε = ε′ in [44], andhave R = 1.

For the born cross-section we find

σb = Abg2

16π(92)

(93)

There are two virtual diagrams, once interfere with the tree lever, givethe next order correction in coupling constant. The first one is the vertexcorrection shown in Fig. 9.

23

Q

2

1

Figure 9: Vertex correction.

We can write the amplitude as

Mv1 =−ig3

(4π)2Γ(3− d/2)

Q2(4π

Q2)2−d/2

∫dzdy

1

(−yz)3−d/2(94)

where d = 4− ε, but ε < 0 so that the integral converges.We have

δM2 =M0M∗v1 +M∗

0Mv1 (95)

= +g4

(4π)22Γ(3− d/2)

Q2(4π

Q2)2−d/2

∫dzdy

1

(−yz)3−d/2(96)

The phase-space integral in d dimension is∫Π2 = (

4π

Q2)2−d/2

1

2d√πΓ(d−1

2)

(97)

We find

σv1 =Ab1

2

∫Π|δMv1|2 (98)

=Abg4

128π3Q2(4πe−γE

Q2)4−d

(− 4

ε2− 4

ε− 4 +

5π2

6

)(99)

where γE is the Euler-Mascheroni constant. The other loop diagram is thevacuum diagram for the intermediate Higgs shown in Fig. 10.

The integral for this diagram is UV divergent but not IR divergent (unlikethe vacuum diagrams of the final particles once we set p21 = p22 = 0). Theamplitude is

Mv2 =ig3

2(4π)2Q2

(4π

Q2

)2−d/2Γ(2− d/2)

∫dx

(x(x− 1))2−d/2(100)

24

Q

1

2

Figure 10: Vacuum correction.

where the factor of two in the first line is the symmetry factor. We have

σv2 =Abg4

128π3Q2

(4πe−γE

Q2

)4−d(− 1

ε− 2)

(101)

Altogether, the cross-section is

σv = σv1 + σv2 = Abg4

128π3Q2

(4πe−γE

Q2

)4−d(− 4

ε2− 5

ε− 6 +

5π2

6

)(102)

Q 3 1

2

+Q

1

2

3

+Q

1

3

2

Figure 11: The real emission.

For the real emission, we have three diagrams, shown in Fig. 11. Theamplitude is given by

Mr =−ig2

Q2

(−1 + x1x2 + (2− x1 − x2)(x1 + x2)

(1− x1)(1− x2)(x1 + x2 − 1)

)(103)

where xi = 2Ei

Q, and x1 + x2 + x3 = 2. The 3-body phase-space integral

in dimensional regularization is∫dΠ3 =

(Q2

4π

)d−4 Q2

128π3Γ(d− 2)×∫ 1

0

dx1

∫ 1

1−x1dx2

1

((1− x1)(1− x2)(x1 + x2 − 1))2−d/2(104)

25

A nice way of doing the integral is by changing the variable x2 = 1 − yx1,0 < y < 1. We find

σr =Ab1

3!

∫dΠ3|MR|2 (105)

=Abg4

128π3Q2

(4πe−γE

Q2

)4−d( 4

ε2+

5

ε+

9

2− 5π2

6

)(106)

The sum of the to two contributions to the total cross-sections is

σv + σr =Abg4

128π3Q2

(4πe−γE

Q2

)4−d(− 3

2

)(107)

Hence, after setting d = 4,

σtotal = Abg2

16π

(1− 3g2

16π2Q2

), (108)

which is a finite number.Although we have used dimensional regularization, but in the massive

theory, in the limit m→ 0, the total cross-section should be the same (108).The limit of integration of the 3-body phase-space (104) is given by theexpressions below:

xmax2 = 1 +b

x1(1− x1)+O(b2) (109)

xmin2 = 1− x1 − 3b+b

x1(1− x1)+O(b2) (110)

xmax1 = 1− 4b (111)

xmin1 = 2√b− b (112)

These will specify the region covered with the blob (Dalitz diagram) asshown in Fig. 12

26

0.2 0.4 0.6 0.8 1.0x1

0.2

0.4

0.6

0.8

1.0

x2

Figure 12: Dalitz diagram for the 3-body decay in massive φ3 theory. Theinside of the black blob is the allowed region. As. m→ 0, the blob will covermore of the upper triangle. The corners of this triangle correspond to softregions, and the sides to collinear regions.

We will skip the calculation here. We point out that in the massive theory,we have learned that in the MS scheme, the mass singularity comes from theresidue of the mass pole. It is given by

R−1 = 1 + Π′(m2p) (113)

where as usual prime is differentiation with respect to p2 and we have

Π =g2

(4π)2Γ(ε/2)

∫ 1

0

dx1

(m2 + p2x(x− 1))ε/2(114)

with d = 4− ε. R is UV finite and we can set ε = 0. We find that

R−1 = 1 +g2

(4π)2

−1 + 2π3√3

m2(115)

3.4.2 AP Function and The Sudakov Form Factor

The calculation of AP function is the same as in the previous sections. Thesplitting shown in Fig. 2 gives

27

1

2!

∫g2

(p2a −m2)2d4pbd

4pc. (116)

Writing the integral over in terms of the θ and z. We have

g2

(4π)2

∫dz

∫ δ

0

dθθ

Q2z(1− z)(θ2 +m2/Q2f(z)

)2 , (117)

where δ is our angular resolution, and

f(z) =1− z + z2

(z − z2)2. (118)

Integrating over the θ, we find

1

2z(1− z)

(1

m2f(z)− 1

Q2δ2 +m2f(z)

)(119)

At this point we need to choose whether δ2 is less than or bigger than m2/Q2.If we fix δ2 < m2/Q2, the expression is finite as δ → 0. In fact this termvanishes for δ = 0, which means that these diagrams do not contribute tothe jets with fewer legs.

However, if we take the limit m2/Q2 → 0 and δ → 0, while keepingδ2 > m2/Q2, the above expression would become singular both in m2/Q2

and δ2. Integrating over z first and then expanding (119) in m2/Q2, we find

g2

2(4π)2

[(− 1 +

2π

3√

3

) 1

m2+

1

Q2δ2+O(m2)

]. (120)

As expected, two times this expression (for each leg) cancels the m2 di-vergence of R in expression (115).

Following our arguments of the previous sections, we want to write theAP function by setting mass to zero and regulated the integral of t withmax[δ2,m2/Q2]. However, we should be careful since we cannot set m = 0 in(117). It gives a wrong result since f(z) has poles at z = 0, 1. We can findthe AP function by changing the variable to kT , it gives (see appendix B foranother derivation)

P(z, t) =g2

(4π)2z(1− z)(

t+m2(1− z + z2))2 (121)

(122)

28

Now, setting m = 0 we find

P =g2

(4π)2z(1− z)

t2(123)

(124)

Our result matches the result in [47]. The power divergence, called ultra-collinear divergence, has been discussed in that paper.

Sudakov form factor is the exponentiation of (123) [44]:

∆(t, t0) = exp

[−∫ t

t0

dt′∫dzP(z, t′)

]≈ exp

[− g2

6(4π)2t0.

]φ3 in 4D (125)

3.4.3 Jet Generating Function

The equation for the generating function does not change with the dimension.Hence, equation (53) for the cubic theory in six dimensions is good here aswell. We again find (57):

Φ(3)[t] =u

u+ (1− u)∆−1(126)

with Sudakov factor given by (125). All the analysis follows as in sixdimensions.

3.5 The Broken Theory In Four Spacetime Dimension

Now, let us consider the presence of both cubic and quartic interactions.

L =1

2

(∂φ)2 − 1

2m2φ2 − g

3!φ3 − λ

4!φ4 (127)

We have shown that the IR divergences for each coupling cancel indepen-dently and there. The Sudakov form factor, by definition, is the multiplica-tion of the Sudakov form factors for the cubic and quartic couplings,

∆(t, t0) =∆φ3(t, t0)∆φ4(t, t0) (128)

= exp

[− g2

6(4π)2t0− λ2

24(4π)4log

t

t0

]. (129)

29

If we look at the Sudakov factor as a function of t0, we can see that aswe decrease t0 the cubic term dominates. But we know that at some pointt0/t drops below the resolution angle δ2 and we need to swap t0 with tδ2.The cubic term becomes proportional to 1/t and quartic term becomes aconstant (a function of δ2). Hence, either as a function of t0 or t, as t0/tdecrease, eventually the quartic term dominates. However, because of thelog divergence vs power divergence, and because of the extra 1/(4π)2, thishappens at an extremely small t0/t. For all realistic purposes, the cubic termis the dominant one which tells us to a good approximation we can ignorethe quartic term in the Sudakov factor and also the quartic splitting.

Turning to the broke theory in 4D, we have

L =1

2(∂h)2 − 1

2m2hh

2 −√λh2mhh

3 − 1

4λhh

4 (130)

And we have for the Sudakov factor

∆h(t, t0) = ∆φ3∆φ4

= exp

[− 3λhm

2h

(4π)2t0− 3λ2h

2(4π)4log

t

t0

]broken φ4 in 4D (131)

The generating function is given by

Φh[t] =∆h(t, t0)Φh[t0]

+ ∆φ4(t, t0)

∫ t

t0

dt′∆φ3(t, t′)

∫dzPφ3(z, t′)Φh[z

2t′]Φh[(1− z)2t′]

+ ∆φ3(t, t0)

∫dx

∫dz

∫ t

t0

dt′∆φ4(t, t′)Pφ4(x, z, t′)×

Φh[x2t′]Φh[z

2t′]Φh[(1− x− z)2t′] (132)

As discussed above, we will ignore the quartic interaction and only con-sider 1→ 2 splitting (we have checked this approximation numerically). Thegenerating function is given by,

Φh[t] ≈ ∆h(t, t0)Φh[t0] +

∫ t

t0

dt′∆h(t, t′)

∫dzPh(z, t′)Φh[z

2t′]Φh[(1− z)2t′]

(133)The solution is identical to the cubic theory in six dimensions (57) with

substitution of the sudakov factor.

30

4 ◦ Off-sehll φ∗ → nφ Process.

The generating function corresponds to the evolution of a highly relativisticand approximately on-shell particle. Hence, to apply this method to theprocess φ∗ → nφ, where the φ∗ is highly off-shell, we approximate the JGFfor this process as

Φφ∗→nφ[t] = Φ(3)[t/4]2 + . . . , (134)

for the cubic theory and

Φφ∗→nφ[t] = Φ(4)[t/9]3 + . . . , (135)

for the quartic theory. The next terms will be of order O(λ2) and higher withadditional sudakov factor. The last expression also works for the un-brokentheory since the 2-body cross section for this theory is suppressed comparedto the 3-body cross-section.

� �� ��� ��� ����

����

����

����

����

����

�(�)

Figure 13: f(n) in equation (137).

For the cubic and quartic theories, the jet rate becomes

R(3)n = (n− 1)∆2(1−∆)n−2, (136)

R(4)n = f(n)n∆3(1−∆2)

n−32 , (137)

with f(n) given in Fig. 13 .

31

�=�

�=�

�=�

�=�

� � � �-���[Δ]

���

���

���

���

���

��

(a) The cubic theory.

�=�

�=�

�=�

�=�

� � � �-���[Δ]

���

���

���

���

���

��

(b) The quartic theory.

Figure 14: The jet rates.

In Fig. 14, we have plotted few of the jet rates as a function of − log[∆].We can see that the multi-scalar rates start to dominate when

− log[∆] > 1 (138)

32

φ3 φ4 Broken Theoryn # of diagrams n # of diagrams n # of diagrams2 1 3 1 2 13 3 5 10 3 44 15 7 280 4 255 105 9 15400 5 2206 945 6 24857 10395 7

Table 1: Number of Feynman diagrams for the process gg → φ∗ → nφ fordifferent theories.

Q2

m2> exp

[12(4π)3

g2]. (139)

This is an extremely large number for any perturbative value of the couplingconstant. Similar situation holds for the other theories. Hence, almost alwayswe will see two jets in any high energy process involving only scalar particles.

5 ◦ Comparison to Fix-order Calculation

For an arbitrary jet clustering algorithm, the divergent logs of the IR pa-rameter does not necessarily exponentiate. In QCD the JADE algorithm isone example [50]. As explained earlier, in QCD it has been proposed andchecked numerically and experimentally that in the KT -Algorithm, the onewe used here, the divergent logs do exponent.

We have further argued that we can change the resolution parameter tothe mass. In this case the jet rates become the particle cross-sections and wecan utilities MADGRAPH [49] to compute cross-section of gg → φ∗ →fewφ. We can expand the expressions for the cross-sections (136) and (137) tofind the leading contributions. Since

∑σn/σ/total = 1, we know that,

σ2/σtotal = 1− σ3/σtotal − σ4/σtotal − . . . , (140)

for the cubic theory (for the quartic theory we have to start from σ3). Sinceσ2/σtotal is proportional to the second power of the sudakov factor, the leadingterms of the cross-sections confirm the exponentiation of these logs. Note that

33

we interpret these logs in σ2/σtotal to come from the virtual corrections, andwe shall not take them into account in our comparison to Madgraph sinceprogram computes the diagrams at tree-level

�=�

�=�

�=�

��-� ��-� ��-� ���� � ���

��-��

��-�

��-�

���

������

� [���]

σ�/σ�

Figure 15: Madgraph computation vs. Jet calculation of gg → φ∗ → nφ inthe φ4 theory. From top to bottom: n = 3,5, and 7.

Another point we shall make is that we expect the total cross-sectionsbecomes independent of mass in the mass-less limit. We have shown this tothe next leading order in all the theories above. To the first order, the totalcross-section is equal to σ2.

�=�

�=�

�=�

�=�

��-� ���� � ���

��-��

��-�

���

����

� [���]

σ�/σ�

(a) g = 1/8

�=�

�=�

�=�

�=�

��-� ���� � ���

��-��

��-�

���

����

� [���]

σ�/σ�

(b) g = 1/80

Figure 16: Madgraph computation vs. Jet calculation of gg → φ∗ → nφ inthe φ3 theory.

34

�=�

�=�

�=�

�=�

��-� ���� � ���

��-��

��-�

���

����

� [���]

σ�/σ�

Figure 17: Madgraph computation vs. Jet calculation of gg → φ∗ → nφ inthe φ3 theory with the inclusion of sudakov factors. g = 1/8

We will use the model HEFT and turn off cubic interactions for analyzingφ4 theory and, vise versa, turn off the quartic coupling to study φ3 theory.Due to the huge number of Feynman diagram, listed in Table. 1, it is notpossible7 to compute more than few particles in the final state.

In Fig. 15, we have computed the cross-section as a function of the scalarmass while fixing the center-of-mass energy at 10 TeV. We can clearly seethat our approximation becomes better at smaller m. While the number ofthe Feynman diagram grow factorially, our result matches the perturbationcalculation.

In Fig. 16, we have repeated the calculation for the cubic theory (noquartic interaction). We again see that the approximation becomes betterat smaller mass. However, here we see that for g = 1/8, at around m = 0.01GeV, the predication fails. At this point the ratio g2/(4π)2m2 becomes largerthan 1, and we cannot trust our prediction at fix order. In Fig. 16b we haveshown that this is the case for g = 1/80 as well. If we do add the sudakovfactors, all the cross-sections will decrease rapidly at these points, as shownin Fig. 17 .

7In a reasonable amount of time using a home computer.

35

6 ◦ Conclusion and Discussion

In summary, we have shown that the IR divergences cancel in scalars theories,permitting defining scalar jets. We found that there is only a single logarithmdivergence in these theories corresponding to collinear divergent. Hence, wedefined scalar jets with the opening angle δ. We argued that for the massivetheory, if δ is smaller than m/Q, where Q is the hard scale, we can substituteδ with m/Q in the Altarelli-Parisi function and consequently in the Sudakovfactor.

We used the Sudakov factor to write an equation for the generating func-tion of jet rates. The jet rates are given by equations (58) and (85), for thecubic theory and the quartic theory respectively. For the broken theory, weargued that the generating function is approximately the same as the cubictheory generating function.

The jet cross-sections are given by σn = Rnσtotal, where the total cross-section corresponds to fix Q (note that

∑nRn = 1). We do not know what

the total cross-section is, but for the purpose of comparing to the past results,it is enough to know the ratios of the jet cross-sections,

σn+1

σn=Rn+1

Rn

. (141)

Since our formalism is based on highly relativistic particles, however, weare not allowed to compare to the threshold result. If we set

Q = nm(1 + ε) (142)

where ε is the average kinetic energy of the final particles, the (??), forexample, is valid in the limits ε � 1 and Q/m � 18 , while the thresholdlimit is at ε� 1.

Nevertheless, we can state few things regarding multi-scalar cross-sections:

• Our result does not support the Holy-grail function. There is no de-pendence on nλ, but on n and λ.

• Our result is in contradiction with the multi-Higgs proposal whichstates that the cross-section will become unsuppressed at high energy,specially since based on their work, the region where the final Higgswill become semi-relativist (ε ≈ 10) becomes unsuppressed as well.

8This is especially important in a massive theory since the splittings that lead to pro-duction of massive particles can become suppressed

36

• A jet contains all the particles that can be fitted into the jet cone andjet energy. The fact that even the two jet cross-section is finite asQ→∞, indicates that the factorial divergence does not exist.

• We have found that while the final particles are relativistic, the cross-sections do not grow factorially. It is thus hard to believe that thedependence of the cross-section on the number of the final particleschange as we change the energy of the particles. It is interesting to seewhy the semi-classical methods sometimes give the factorial divergentas well. It is also interesting to see why in the calculations that sumthe leading virtual corrections to all order, the logm/Q does not showup.

ACKNOWLEDGMENT: This work was the continuation of part of myPhD thesis, done in the University of California in Davis, under the super-vision of Dr. John Terning. Without his guidance and help this work wouldnot have been possible. I thank Miranda Chen for sharing her calculationwith me. I thank Fayez Abu-Ajamieh, with whom the early stages of thiswork was developed. I am also thankful to Dr. Mahdi Torabian for hostingme in the Sharif University. I was supported by Bonyad-e Meli-e NokhbeganFund.

37

A ◦ Loop Corrections to All Order at Threshold for φ3

Theory in Six Spacetime Dimensions

The one-loop correction to multi-scalar process in the cubic theory can befind using the methods in [6, 10, 11].

Let us define ai(n) as the amplitude of producing n particle at i’th looporder We have

φ0(x) =∑ ia0(n)xn

n!(n2 − 1)(143)

φ1(x) =∑ ia1(n)xn

n!(n2 − 1)(144)

We start by writing the one loop amplitude recursively via Fig. 18. Wehave

a1(n)

n!=− ig

∑n1,n2

ia1(n1)

n1!(n21 − 1)

ia0(n2)

n22(n

22 − 1)

− ig2

∫dDk

(2π)DD(n, k)

n!

− iT2ia0(n)

n!(n2 − 1)

− iT3ia0(n1)

n1!(n21 − 1)

ia0(n2)

n2!(n22 − 1)

(145)

= +

+ +

Figure 18: The recursion relation for the one loop corrections.

Where q = (1, 0, 0, 0) and we have set the mass equal to unity in thedenominators. The T3 is the mass counter-term and the T2 is the mass and

38

field counter-terms:

− iT2 = −i(m2δm + n2m2δZ

),

− iT3 = −iδg,

where from the usual renormalization we know that

δm =− 1

ε

g2

(4π)2+ (146)

δZ =1

6ε

g2

(4π)2+ (147)

δg =− 1

ε

g3

(4π)3(148)

The cooeficient of the δZ term is p2 = (nm)2. Let us also define,

f(x, k) =∑ −iD(n, k)xn

n!. (149)

Multiplying (145) by xn and summing over x, we find an equation for φ1:(xd

dx

(xd

dx

)− 1)φ1(x) =gφ1(x)φ0(x) +

g

2

∫dDk

(2π)Df(x, k) (150)

+m2[δmφ0(x) + δZ

(φ0(x) +

g

2φ20

)](151)

+g

2δgφ

20 (152)

The recursion relation for the propagator is given in Fig. 19 and is

D(n, k)

n!=− ig i

(k + nq)2 − 1 + iε

∑ ia(n2)

n2!(n22 − 1)

D(n1, k)

n1!(153)

Writing x = −12geτ and f(x,K) = ye−ετ and ε = kq = k0 and ω =√

~k2 + 1− iε, the equation for y would be( d2dτ 2− ω2 +

3

cosh τ/2

)y = eετ (154)

39

=

Figure 19: The recursion relation for the propagator.

With u = eτ , the solutions are

f1 =(u−ω(1 + u)−3

)(3− 27u+ 27u2 − 3u3 − 11ω + 27uω + 27u2ω

− 11u3ω + 12ω2 + 12uω2 − 12u2ω2 − 12u3ω2

− 4ω3 − 12uω3 − 12u2ω3 − 4u3ω3)

f2 =f1(ω → −ω)

The Wronskian is W = 2ω(9− 49ω2 + 56ω4 − 16ω6). The solution for f is

f(x,K) =e−ετ

W

(f1

∫dseεsf2 + f2

∫dseεsf2

). (155)

The K0 integral in (150) gives a delta function and we find

g

2

∫dDk

(2π)Df(x, k) =

g

2

∫ddK

(2π)df1f2W

, (156)

where d = D − 1. It is possible to write this expression entirely in terms ofφ0. We have

g

2

∫ddK

(2π)d

[ 1

2√

1 + ~K2

][1

2− gφ0

3 + 4 ~K2+

5g2φ20

(24 + 32 ~K2) ~K2(157)

+25g3φ2

0

(15 + 8 ~K2 − 16 ~K4)24 ~K2

](158)

The first integral gives exactly the tadpole contribution. The divergentparts of the second and third terms are

g

2

∫ddK

(2π)d1

2√

1 + ~K2

−g2φ0

6 + 8 ~K2=

g2

(4π)35

6ε+ const. (159)

g

2

∫ddK

(2π)d1

2√

1 + ~K2

g3φ20

(24 + 32 ~K2) ~K2=

g3

(4π)35

12ε+ const., (160)

40

where d = 6−ε and A and B are some numerical constant. The divergentof the first integral is canceled by the O(φ0) part of the counter-terms in(151). And the divergent of the second integral is canceled by the remainingpart of δZ and δg.

The equation for φ1 bowls down to(xd

dx

(xd

dx

)− 1− gφ0

)φ1(x) = Ag2φ0(x) +Bg3φ0(x)2Cg

4φ30. (161)

It is customery to set the finite part of the counter-terms such A and Bare zero [Agyres, Smitt]. We will do the same but we should warn that thesolution with these coefficients might lead to φ1 = log x+ . . . terms that aresingular at x = 0.

With A = B = 0, we find that

φ1(x) =18g2x

(1− g12x)4

(162)

We finally find the amplitude to the first order

An =dn

dxn(φ0 + φ1)

∣∣x=0

= nn!( g

12

)n−1[1 + 3g2(n2 + 3n+ 2)

](163)

B ◦ AP Function for φ3 Theory in Four Spacetime Di-mensions

a

b

c

Figure 20: φ → φφ. Momentums are flowing to the right. “c” is integratedover.

We start by writing the cross-section for production of n+ 1 particles interms of the cross-section of production of n particle plus a split particles,labeled c (Fig. 20), that is radiated from one of the final legs. We have

41

σ1→n+c = flux factor× 1

n!

∫dΠfdΠc |Mn+c|2 (164)

= flux factor× n

n!

∫dΠfdzdk2TP(z, kT )|Mn|2 (165)

= σ1→n

∫dzdk2TPφ3(z, kT ) (166)

where c is assumed to be collinear to the leg labeled b. We have definedz = Eb/Ec. The P is the splitting function 9. The n in the second linecomes from the fact that the n+ 1 phase-spaces decomposes to n regions, ineach region c is collinear to one of the n legs, and each region gives the samefactor.

We write the momentum as (called Sudakov decomposition [30]),

pb = zpa + βn+ kT (167)

pc = (1− z)pa − βn− kT (168)

so that pa = pb + pc. The vector n is an arbitrary vector perpendicular tokT , we choose it to be (1, 0, 0,−1). The phase-space integral for particle cbecomes

dΠc =dzkTdkTdφdβ

(2π)4× J(= Ea + pa3)× (2π)δ(p2c −m2) (169)

=dzdk2T

4(2π)2(1− z)(170)

where in the second line we have integrated over β and φ. We further have

p2a = p2b + p2c + 2pbpc =k2t

z(1− z)+

p2b(1− z)

+p2cz, k2T , p

2b,c � Eb,c (171)

Hence,

Pφ3 =g2

4(2π)21

z(1− z)

1

(p2a −m2)2. (172)

The extra 1/z in the first fraction comes from changing the phase-space factor

9Contrary to what is usual, we have moved the kT dependence into the definition of Psince in the case of massive particles it does not factor out.

42

of particle b to that of particle a (Fig. 20). Using (171) we find

Pφ3 =g2

16π2

z(1− z)(k2T + zp2b + (1− z)p2c − z(1− z)m2

)2 (173)

=g2

16π2

z(1− z)(k2T +m2(1− z + z2)

)2 (174)

(175)

C ◦ AP Function for φ4 Theory in Four Spacetime Di-mensions

The cross-section is

σ1→n+c+d = flux factor×∫dΠfdΠcdΠd |Mn+c+d|2 (176)

= flux factor×∫dΠfdzdxdk2Tcdk

2TdPφ4(z, x, kTc , kTd)|Mn|2 (177)

= σ1→n

∫dzdxdk2Tcdk

2TdPφ4(z, x, kTc , kTd) (178)

where z = Ec/Ea and x = Ed/Ea. The Sudakov decomposition of momentumis given by

pb = (1− z − x)pa + (β + α)n+KTc −KTd (179)

pc = zpa − αn−KTc (180)

pd = xpa − βn+KTd (181)

so that pa = pb+pc+pd. The vector n is chosen such that it is perpendicularto KTc,d and n2 = 0. Choosing it to be (1,0,0,-1), gives n.pa ≈ 2Ea, assumingthat pa is highly boosted. The phase-space integrals become

dΠbdΠcdΠd = dΠadzdz′dk2Tcdk

2Tddφcdφd

16(2π)6E2azx(1− z − x)

(182)

We can find β and α by imposing on-shell condition for particles c and d. Assuming that all the particles are massless, these conditions give

2αn.pa = zp2a −K2Tc/z and 2βn.pa = xp2a −K2

Td/x (183)

43

Using these equations and on-shellness of the final particles, we arrive at

p2a =1

zx(1− z − x)

[z(1− z)k2Tc + x(1− x)k2Td − 2zxkTckTd cosφ

](184)

where φ is the angel between KTc and KTd . Noting that Mn+c+d = λp2aMn,

and furthermore integrating over the azimuthal angels, we find the splittingfunction to be

Pφ4(z, x, kTc , kTd) =λ2

16(2π)4zx(1− z − x)[z(1− z)k2Tc + x(1− x)k2Td ][

(z(1− z)k2Tc + x(1− x)k2Td)2 − 4z2x2k2Tck2Td

]3/2(185)

44

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