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Theoretical Physics Institute

arXiv:hep-ph/9209240v1 15 Sep 1992Theoretical Physics InstituteUniversity of MinnesotaTPI-MINN-92/45-TSeptember 1992Summing one-loop graphs at multi-particlethresholdM.B. VoloshinTheoretical Physics Institute, University of MinnesotaMinneapoilis, MN 55455andInstitute of Theoretical and Experimental PhysicsMoscow, 117259AbstractIt is shown that the technique recently suggested by Lowell Brown for summing thetree graphs at threshold can be extended to calculate the loop effects.

Explicit resultis derived for the sum of one-loop graphs for the amplitude of threshold productionof n on-mass-shell particles by one virtual in the unbroken λφ4 theory. It is alsofound that the tree-level amplitude of production of n particles by two incomingon-mass-shell particles vanishes at the threshold for n > 4.

The problem of calculating amplitudes of processes with many weakly interact-ing particles has recently attracted a considerable interest, initially triggered bythe observation[1, 2, 3] that such processes in particular are associated with a pos-sible baryon and lepton number violation in high-energy electroweak interactions.Cornwall[4] and Goldberg[5] have pointed out that in perturbative amplitudes withmany external particles the weak coupling may get compensated by large number ofdiagrams. This is a manifestation of the old-standing problem of the factorial growthof the coefficients in the perturbation theory[6].

Since the perturbative expansion formulti-particle amplitudes starts from a high order in the coupling constant, for suffi-ciently large number n of particles the factorial growth of the coefficients in the seriesinvalidates the perturbative calculation of such amplitudes. Given lack of a betterapproach it seems useful to quantify and study the problem within the perturbationtheory itself.

A simple model example in which the problem arises with full strengthis the amplitude An where a virtual particle of a real scalar field φ produces a largenumber n of on-mass-shell φ-particles in the λφ4 theory. It has been recently foundthat the sum of all tree graphs for this amplitude in the threshold limit, i.e.

when allthe produced particles are at rest, can be calculated exactly for arbitrary n both inthe case of unbroken symmetry[7] and in the case of theory with spontaneous breakingof the symmetry under the reflection φ →−φ [8].Originally the calculation[7] was done by directly solving a recursion relation forthe tree graphs. Argyres, Kleiss and Papadopoulos[8] applied a regular method ofsolving the recursion relations based on generating function for which the recursionrelation for the amplitudes An is equivalent to second order non-linear differentialequation.

Most recently Brown[9] has shown that the generating function is nothingelse than the classical field φ0(t) generated by an external source ρ = ρ0eimt which fieldis a complex solution of the Euler-Lagrange classical equation satisfying the conditionthat it has only the positive frequency part. The equations and their solutions in bothapproaches are related by a simple change of variable.

Thus Brown has reproducedthe previous results[7, 8] in a simple and elegant way.The purpose of this paper it to show that Brown’s technique can be extended tocalculate the loop contributions to the amplitudes An as well as the amplitudes ofmore complicated processes e.g. of the scattering 2 →n.

For definiteness the case ofunbroken λφ4 theory with the Lagrangian1

L = 12(∂φ)2 −12m2φ2 −14λφ4(1)will be considered and we will concentrate on the calculation of the one-loop correctionto the threshold amplitudes An. The result of this calculation can be written as⟨2k + 1|φ(0)|0⟩= (2k + 1)!

¯λ8 ¯m2!k "1 −k(k −1)33/2λ16π2 ln 2 +√32 −√3 −i π!#,(2)where ¯λ and ¯m are the renormalized coupling constant and the mass, the appropriaterenormalization condition, which specifies the finite terms, will be described below.Equation (2) gives the exact sum of all tree and one-loop graphs at the thresholdof production of n = 2k + 1 φ-particles for arbitrary k 1. The formula (2) givesthe relative magnitude of the first loop correction growing as n2 at large n. Thisbehavior explicitly demonstrates invalidity of the previous arguments[10, 11] of thepresent author that terms, containing n2λ should be absent in the loop effects.

Thereason for those arguments to be faulty is related to the singularities of the underlyingclassical field in the plane of complex time, where the quantum fluctuations are moresingular than the classical background. However, the presence of the n2λ parameterin the loop effects does not necessarily imply that the quantum effects completelyeliminate the growth of the amplitudes and a further study is needed.

The relativelyeasy calculability of the threshold amplitudes at the tree and the one-loop level andthe remarkably simple form of the result (2) gives us a hope that these amplitudescan be studied well beyond the first terms in the perturbative expansion.That the amplitudes in the λφ4 theory at the multi-particle thresholds may havespecial properties is also hinted at by another fact, which follows as a by-product fromthe calculation in this paper. Namely, if one considers the sum of all tree graphs forthe amplitude of the process where two incoming on-shell particles produce n on-shellparticles exactly at the threshold, it turns out that this sum is non-zero only for n = 2and n = 4 and is vanishing for all n > 4 (the number of final particles in this case isnecessarily even).

It is in fact due to this behavior that the one-loop term in eq. (2)contains only the 4-particle threshold factor.The technique suggested by Brown[9] is based on the standard reduction formula1The number n of final particles produced by one virtual is necessarily odd due to the unbrokenreflection symmetry.2

representation of the amplitude through the response of the system to an externalsource ρ(x), which enters the term ρφ added to the Lagrangian⟨n|φ(x)|0⟩=" nYa=1limp2a→m2Zd4xa eipaxa(m2 −p2a)δδρ(xa)#⟨0out|φ(x)|0in⟩ρ|ρ=0 ,(3)the tree-level amplitude being generated by the response in the classical approxima-tion, i.e. by the classical solution φ0(x) of the field equations in the presence of thesource.For all the spatial momenta of the final particles equal to zero it is sufficient toconsider the response to a spatially uniform time-dependent source ρ(t) = ρ0(ω)eiωtand take the on-mass-shell limit in eq.

(3) by tending ω to m. The spatial integralsin eq. (3) then give the usual factors with the normalization spatial volume, which asusual is set to one, while the time dependence on one common frequency ω impliesthat the propagator factors and the functional derivatives enter in the combination(m2 −p2a)δδρ(xa) →(m2 −ω2)δδρ(t) =δδz(t) ,(4)wherez(t) =ρ0(ω)eiωtm2 −iǫ −ω2(5)coincides with the response of the field to the external source in the limit of absenceof the interaction, i.e.

of λ = 0. For a finite amplitude ρ0 of the source the responsez(t) is singular in the limit ω →m.

The crucial observation of Brown[9] is that, sinceaccording to eq. (4) we need the dependence of the response of the interacting field φonly in terms of z(t), one can take the limit ρ0(ω) →0 simultaneously with ω →min such a way that z(t) is finite,z(t) →z0eimt .

(6)Furthermore, to find the classical solution φ0(x) in this limit one does not haveto go through this limiting procedure, but rather consider directly the on-shell limitwith vanishing source. The field equation with zero source is of course given by∂2φ + m2φ + λφ3 = 0 .

(7)3

For the purpose of calculating the matrix element in eq. (3) at the threshold onelooks for a solution of this equation which depends only on time and contains onlythe positive frequency part with all harmonics being multiples of eimt.

The solutionsatisfying these conditions reads as[9]:φ0(t) =z(t)1 −(λ/8m2)z(t)2(8)According to equations (4) and (3) the n-th derivative of this solution with respectto z gives the matrix element ⟨n|φ(0)|0⟩at the threshold in the tree approximation:⟨2k + 1|φ(0)|0⟩0 = ∂∂z!2k+1φ0|z=0 = (2k + 1)! λ8m2!k,(9)which reproduces the previously known result[7].

The fact that the matrix elementis non-zero only for odd n obviously follows from that the expansion of φ0 in eq. (8)contains only odd powers of z.It can be noticed that the solution (8) is in fact not uniquely determined by theabove mentioned conditions.

Namely, z(t) can be rescaled by an arbitrary constantC. This constant corresponds to the choice of normalization of the field, so that thevalue C = 1 is fixed by the usual normalization condition ⟨1|φ(0)|0⟩= 1, as can beseen from the linear term in the expansion of φ0 in powers of z.Another important point concerning the solution (8) is related to the fact thatthis solution is essentially complex for real time t. This is imposed by the that incalculating production of particles by the virtual field, rather than both productionand absorption, one necessarily has to consider only the positive frequency part ofthe field, which is essentially complex.The quantum loop corrections to the amplitudes ⟨n|φ|0⟩are obtained by substi-tuting instead of the classical field the mean value of the full fieldφ(x) = φ0(x) + φq(x) ,(10)where φq(x) is the quantum part of the field.

Expanding the field equation (7) near theclassical solution φ0 and retaining only the first non-vanishing quantum correction,one finds that the mean field φ(x) to the first quantum order satisfies the equation∂2φ(x) + m2φ(x) + λφ(x)3 + 3λφ0(x)⟨φq(x)φq(x)⟩= 0 ,(11)4

where ⟨φq(x)φq(x)⟩is the limit of the Green function in the classical background fieldφ0G(x1, x2) = ⟨T(φq(x1)φq(x2))⟩(12)when its arguments are at the same point x.Therefore the steps needed to calculate the first loop correction to the amplitudesAn are the following:i. Calculate the Green function (12) as the inverse of the operator of the secondvariation of the action,∂2 + m2 + 3λφ0(x)2 ;(13)ii.

Find its limit in coinciding points, which enters eq.(11);iii. Expand the solution of thus found equation in powers of z(t), which gives theamplitudes at the threshold in the same way as in eq.

(9).This program however is obscured at the very first step by the fact that with es-sentially complex φ0 (eq. (8)) the operator (13) is essentially non-Hermitean.

Howeverone can render this operator real and thus the problem more tractable by analyticalcontinuation in time t, which amounts to rotation and shift in the complex plane.Namely, the substitution which achieves the goal reads assλ8m2z(t) = iemτ(14)and the variableτ = it + 1m ln λz08m2 −iπ2m(15)is then used as the new time variable t. The necessity of the shift in addition to theusual rotation to the Euclidean time is caused by existence of a pole of φ0(t) on thenegative imaginary axis, where the operator (13) is singular. The poles are repeatedparallel to the real axis with the period π/m.

The axis, corresponding to real τ,on which the operator (13) is real runs parallel to the imaginary axis of t exactlyin the middle between two poles, see Fig.1. It should be emphasized however thatit is the pole structure of the field, which gives rise to the factorial growth of themulti-particle amplitudes the study of which may eventually be the central point insolving the problem of multi-boson processes.

Here, for the purpose of the specific5

calculation, we chose to stay away from the poles to avoid explicit singularity in theequations.To somewhat simplify the notation we set the mass m equal to one and restore itwhen needed and also introduce the notation u(τ) = eτ = −iqλ/8z(t). For real u(τ)the classical field (8) is purely imaginary:φ0(u(τ)) =s8λiu1 + u2 =s2λicosh τ(16)and the operator (13) is real.

In a mode with spatial momentum k the operator hasthe form−d2dτ 2 + ω2 −6(cosh τ)2 ,(17)which is the familiar operator in one of exactly solvable potentials in Quantum Me-chanics (see e.g. Ref.

[12]), and ω is the energy of the mode: ω2 = k2 + 1.The regular at τ →+∞solution of the homogeneous equation with the operator(17) has the formf1(u(τ)) = 2 −3 ω + ω2 −8 u2 + 2 ω2 u2 + 2 u4 + 3 ω u4 + ω2 u4uω (1 + u2)2(18)and the solution regular at τ →−∞is given byf2(u(τ)) = f1(1/u(τ)) = uω (2 + 3 ω + ω2 −8 u2 + 2 ω2 u2 + 2 u4 −3 ω u4 + ω2 u4)(1 + u2)2. (19)The Wronskian of these solutions is given byW = f1(τ)f ′2(τ) −f ′1(τ)f2(τ) = 2ω(ω2 −1)(ω2 −4) .

(20)The convention for the sign of the Green function used here is specified by the explicitexpression for the Green function in partial wave with the spatial momentum k interms of f1, f2 and W: Gω(τ1, τ2) = f1(τ1)f2(τ2)/W for τ1 > τ2 and Gω(τ1, τ2) =f1(τ2)f2(τ1)/W for τ2 > τ1.Naturally, having the explicit expression for the Green function one can also eval-uate amplitudes of more complicated processes, say, the tree level amplitude of thethreshold production of n particles by two incoming on-mass-shell particles of high6

energy. However equations (18), (19) and (20) show that there is in fact almost noth-ing to calculate for the latter amplitude: the Green function has poles only at ω2 = 1and ω2 = 4 (the zeros of the Wronskian (20)).

By the reduction formula this impliesthat the on-mass-shell amplitude is non-vanishing only at these values of the energyof each of the two incoming particles. The case ω = 1 corresponds to the trivialprocess 2 →2 at the threshold, while the case ω = 2 corresponds to the process2 →4.

(In the rest frame of the produced particles, which is used throughout thispaper, ω corresponds to the energy of each of the two incoming particles, so that thetotal energy is 2ω = 4.) The absence of other poles of the Green function at higherω means that for n > 4 the sum of tree graphs for the on-shell process 2 →n isvanishing.After this remark we proceed with calculating the loop correction in eq.(2).

Thepartial wave Green function at coinciding points is given bygω(τ) = Gω(τ, τ) = f1(τ)f2(τ)/W(21)which yields the average value of the square of quantum fluctuations after integratingover k:⟨φq(τ)φq(τ)⟩= g(τ) =Zgω(τ) d3k(2π)3 =12π2Z1 gω(τ)ω√ω2 −1 dω . (22)The calculation of the latter integral involves problems related to the on-shell singu-larities and to the ultra violet divergence.

The on-shell singularities correspond tothe zeros of the Wronskian (20) at ω2 = 1 and ω2 = 4. The first of these correspondsto the translational zero mode of the classical solution φ0 and in fact produces noeffect in the integral in eq.

(22) since the singularity at ω2 = 1 is integrable. (Thisis why in the 4-dimensional theory one does not have to consider subtraction of thecontribution of the zero mode from the Green function.) The pole at ω2 = 4m2 (thedependence on mass is restored) is dealt with using the Feynman’s iǫ rule i.e.

byshifting the pole to the negative half-plane m2 →m2 −iǫ. The integral then developsimaginary part, which in the end corresponds to the dynamical imaginary part of theone-loop graphs, dictated by the unitarity.To separate the ultra violet divergent terms we expand gω(τ) in powers of ω−1and find that the two terms, which give the quadratic and the logarithmic divergencehave the form7

gω(τ) = 12ω +6u2(1 + u2)2ω3 + grω(τ) ,(23)where the regular part grω(τ) contains terms of the order ω−5 and higher, so thatits contribution to the integral in eq. (22) is finite in the ultra violet.After thisdecomposition the result of the integration in eq.

(22) can be presented asg(τ) = 12I1 +6u2(1 + u2)2(I3 +12π2) −6u4(1 + u2)4F ,(24)whereF =√32π2 ln 2 +√32 −√3 −i π! (25)and I1, I3 are the ultra violet divergent integrals:In =12π2Zω1−n√ω2 −1 dω .

(26)In equation (24) we have also combined with the logarithmically divergent integrala part of the finite contribution from integration of the grω(τ), which has the samefunctional dependence on u(τ), hence the factor (I3 + 1/(2π2)).The divergent contributions can be regularized in a standard way, the moststraightforward being the Pauli-Villars regularization. Upon substitution into equa-tion (11) for the mean field with the quantum correction the quadratically divergentpart proportional to I1 gives rise to a term linear in the classical field φ0 while the log-arithmically divergent part proportional to I3 results in a correction to the term withλφ30.

Therefore these terms can be dumped into the definition of the renormalizedmass ¯m and the coupling constant ¯λ according to¯m2 = m2 + 3λ2 I1¯λ = λ −9λ24 (I3 +12π2) . (27)These definitions can be used to relate the quantities ¯m and ¯λ to the renormalizedconstants in any other renormalization scheme.

One can readily see that the divergentparts are scheme-independent, while the relation between the finite parts depends onthe specific definition of the regularization procedure.8

The only non-trivial modification of the average field given by equation (11) isrelated to the finite part of the average value of the square of quantum fluctuations(eq. (22)), proportional to the constant factor F.If one seeks the solution of theequation (11) in the form φ(t) = φ0(t; ¯m, ¯λ) + φ1(t), where the renormalization ofthe constants is plugged into the functional dependence of the classical solution, theequation for the correction φ1(τ) (i.e.

on the τ axis) reads as d2dτ 2 −1 +24u2(1 + u2)2!φ1 = −i18λs8λFu5(1 + u2)5 ,(28)the condition on the appropriate solution to this equation being that its expansion inu starts with the fifth power, since only starting from final states with five particles thethreshold amplitudes develop an imaginary part, which in this calculation originatesin the imaginary part of F. The solution satisfying this condition isφ1(τ) = −i3λ4s8λFu5(1 + u2)3 ,(29). Using equation (14) one can readily restore from here the response of the field interms of z(t) with the first quantum correction included:φ0+1(t) =z(t)1 −(¯λ/8 ¯m2)z(t)2 1 −3λ4 F(λ/8m2)2z(t)4(1 −(λ/8m2)z(t)2)2!

(30)and by expanding in series in z(t) finally arrive at the result in equation (2).The rotation (14) used here may invite the objection, that such rotation in thepath integral is obstructed by the infinite chain of poles parallel to the real axis oft, which may give rise to extra contributions in the quantum effects. However it canbe explicitly shown that this does not happen at least at the one-loop level.

Namely,it is a straightforward (but rather cumbersome) exercise to verify that the recursionrelations for the sum of graphs for the propagator of the field φ with emission of non-shell particles all being at rest are equivalent to the differential equation for theGreen function of the operator (17) and then that the recursion relations for the loopgraphs are equivalent to the equation (28) on the τ axis. Another simple (and in noway rigorous) check is to verify the formula (2) for few first n by direct computationof the graphs.

This also turned to be helpful in checking the relative coefficients andsigns in the equations of this paper. The remarkably simple form of the result (2)suggests that there may be a way to calculate further quantum effects.

In particular9

one can notice that the finite term, proportional to the factor F (eq. (25)) has theform given by the simple scalar vacuum polarization at q2 = 16m2.

This of course isa consequence of the eigenmode of the operator (17) at ω2 = 4, or, equivalently, ofthe fact that the tree-level threshold amplitudes of the processes 2 →n are equal tozero for n > 4. In terms of the graphs the cancellation of the contributions to theimaginary and the real parts of the thresholds at higher q2 looks quite surprising.In the present calculation we have avoided approaching the poles of the classicalsolution φ0, where the quantum expansion in fact breaks down, since the quantumfluctuations are more singular than the classical solution.However those are thesingularities of the field in the complex plane of t (or equivalently of z) which giverise to the factorial growth of the amplitudes.

The appearance of the singularity atthe imaginary axis follows from the simple fact that on this axis the classical fieldequationd2dt2φ = m2φ + λφ3(31)corresponds to the free fall in the inverted λφ4 potential which takes a finite time fora finite staring value φ(0). It looks at least extremely unnatural that quantum effectswould slow down this fall to the extent that the time of the fall would be infinite.

Forany finite time, however, the singularity of the field there will produce the factorialgrowth of the amplitudes.As a simple final remark, it can be mentioned that though the present calculationis done for the case of unbroken symmetry it looks quite straightforward to apply thesame technique to the case of the spontaneously broken symmetry.I am thankful to Lowell Brown for sending to me the preprint of his paper andto Arkady Vainshtein for stimulating discussions. I also acknowledge an extensiveuse of the Mathematica[13] software in doing the calculations and preparing the textof the present paper.

This work is supported in part by the DOE grant DE-AC02-83ER40105.References[1] A. Ringwald, Nucl. Phys.

B330, 1 (1990). [2] O. Espinosa, Nucl.

Phys. B343, 310 (1990).10

[3] For a recent review see M.P. Mattis, Phys.

Rep. 214, 159 (1992). [4] J.M.

Cornwall, Phys. Lett.

243B, 271 (1990). [5] H. Goldberg, Phys.

Lett. 246B, 445 (1990).

[6] For a review see e.g. J. Zinn-Justin, Phys.

Rept. 70, 109 (1981).

[7] M.B. Voloshin, Minnesota preprint TPI-MINN-92/1-T, Nucl.

Phys. B, to bepublished.

[8] E.N. Argyres, R.H.P.

Kleiss and C.G. Papadopoulos, CERN preprint CERN-TH-6496 (1992)[9] L.S.

Brown, Univ. Washington preprint UW/PT-92-16.

[10] M.B. Voloshin, Phys.

Rev. D43, 1726 (1991).

[11] M.B. Voloshin, Minnesota preprint TPI-MINN-92/27-T, to appear in Proc.

YaleWorkshop on Baryon Number Violation at High Energy. [12] L.D.

Landau and E.M. Lifshits, Quantum Mechanics, Non-Relativistic Theory,Third edition. Pergamon Press.

[13] Wolfram Research, Inc., Mathematica Version 2.1, Wolfram Research, Inc.,Champaign, Illinois, 1992.11

✉✉✉Im tRe tRe τFig.1 The structure of the classical field φ0(t) (eq. (8)) in the complex tplane.

Heavy dots indicate the poles of φ0. The vertical line going betweenthe poles is the axis of real τ on which the operator (13) is real.12

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