Scattering on the Moduli Space of N=4 Super Yang-Mills

0.18 MB·English

by  Robert M. Schabinger

PreprinttypesetinJHEPstyle-HYPERVERSION
= 4
Scattering on the Moduli Space of
8 N
0 Super Yang-Mills
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RobertM. Schabinger

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DepartmentofPhysics,UniversityofWashington,Seattle,WA 98195-1560
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ABSTRACT: Wecalculate one-loop scattering amplitudesin = 4superYang-Millstheoryaway
[ N
fromtheoriginofthemodulispaceanddemonstrate thattheresultsareextremelysimple,inmuch
1
v thesamewayasintheconformally invariant theory. Specifically, weconsider themodelwherean
2 SU(2) gauge group is spontaneously broken down to U(1). The complete component Lagrange
4
5 density of the model is given in a form useful for perturbative calculations. We argue that the
1 scattering amplitudes with massive external states deserve further study. Finally, our work shows
.
1 that loop corrections can be readily computed in a mass-regulated = 4 theory, which may be
0 N
relevant in trying to connect weak-coupling results with those at strong coupling, as discussed
8
0 recently byAldayandMaldacena.
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X KEYWORDS: NLOComputations, ExtendedSupersymmetry, Spontaneous SymmetryBreaking.
r
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∗[email protected]
Contents
1. Introduction 1
2. Setup 4
3. One-LoopFour-PointScatteringAmplitudes 6
4. SummaryandFutureDirections 10
A. ScalarBoxIntegrals 12
B. TheLagrangeDensity 12
C. DiagrammaticRepresentations oftheScatteringAmplitudes 15
1. Introduction
TheLagrange density of = 4super Yang-Mills theory wasfirstwrittendownlongago[1]and,
N
shortly thereafter, the first one-loop scattering amplitude in the model was calculated [2]. Since
then, such scattering amplitudes have been extensively studied by many groups (see e.g. [3, 4]).
= 4SYM,however,hasnon-trivialdynamics[5]. Thetheorypossessesamodulispaceofvacua
N
parametrized bythevacuumexpectationvalues(VEVs)ofthethreescalarandthreepseudo-scalar
fields in the model. It should be stressed that, in going to a generic point in the moduli space, the
= 4supersymmetry willbepreserved, butthegaugegroup ofthetheory willbespontaneously
N
broken. The theory described in [1] is the conformal phase of = 4 SYM, where all the VEVs
N
areequaltozero. Sofar,scatteringamplitudesawayfromtheoriginofthemodulispace(forstates
intheCoulombphaseofthetheory)havereceivedrelatively littleattention.
= 4 SYM is a very special four dimensional quantum field theory and, consequently, its
N
S-matrix has several unusual properties. We begin by reviewing the interesting features of weak-
coupling perturbation theory in the conformal theory. Thefield content of the model consists of a
gaugefieldA ,fourMajoranafermionsψ ,threerealscalarsX ,andthreerealpseudo-scalars Y .
µ i p q
All fields are in the adjoint representation of a compact gauge group, G. In this work, we choose
G = SU(2) for simplicity. In this case, a generic field, φ, may be written in terms of its color
components asφ = φa σ ,whereσ aretheusualPaulimatrices. TheLagrangedensityof = 4
2 a a N
–1–
isgivenby[6]1
1
= tr F Fµν +ψ¯D/ψ +DµX D X +DµY D Y (1.1)
µν i i p µ p q µ q
L − 2
(cid:26)
+igψ¯αp [X ,ψ ] gψ¯γ βq[Y ,ψ ]
i ij p j − i 5 ij q j
g2
[X ,X ][X ,X ]+[Y ,Y ][Y ,Y ]+2[X ,Y ][X ,Y ] ,
l k l k l k l k l k l k
− 2
(cid:18) (cid:19)(cid:27)
wherethe4 4matricesαp andβq aregivenby2
×
iσ 0 0 σ 0 σ
α1 = 2 , α2 = − 1 , α3 = 3 , (1.2)
0 iσ σ 0 σ 0
2! 1 ! 3 !
−
iσ 0 0 iσ 0 σ
β1 = − 2 , β2 = − 2 , β3 = 0 .
0 iσ iσ 0 σ 0
2! 2 ! 0 !
− −
Oncethegaugegroupandcoupling constant g arefixed,thetheoryisuniquely specified.
It is by now well known that = 4 SYM scattering amplitudes are ultraviolet-finite [8, 9,
N
10, 11, 12]. A modern characterization of UV finiteness in gauge theories with references to the
older literature is found in [13]. In the conformal phase, all the external legs in a given scattering
processaremassless. Ingeneral,one-loopscatteringamplitudesinmasslessquantumfieldtheories
may be written in terms of a basis consisting of certain bubble, triangle, and box scalar Feynman
integrals withclusters ofsubsets ofthenexternal momentaexiting eachvertex[14,15,16,17]. In
= 4SYMadirectcalculation oftheeffectiveactionshowsthatthen-gluonone-loop scattering
N
amplitude canbewrittenintermsofscalarboxFeynmanintegralsonly. [3,18]
This result is somewhat surprising, even in view of the fact that in = 4 SYM scattering
N
amplitudes all UV divergences must cancel. Scalar box and triangle integrals with no internal
masseshavedivergences, butonlyintheinfrared, whereasthescalarbubbleintegralwithnointer-
nalmasseshasonlyanultravioletdivergence. Thus,wewouldnaivelyexpecttheone-loopn-gluon
amplitude tocontainbothboxesandtriangles. Itturnsoutthatthisnaiveexpectation iswrongand
all the scalar triangle integrals cancel out aswell. Byderiving the = 4analog (see [19])ofthe
N
wellknown = 1supersymmetric Wardidentities [20,21,22]andapplying themtothen-gluon
N
scattering amplitudes, itcanbeshownthatamuchlarger classof = 4amplitudes mustbepure
N
scalar box integrals as well. In particular, all four-point and five-point amplitudes fall into this
category.
A natural question is how much of this interesting structure is preserved when we go to the
Coulomb phase of the theory. There the situation is somewhat different because, in the Coulomb
phase, we must distinguish scattering amplitudes which have only massless external states from
those that include some states from massive sector of the theory. Before dicussing this, let us
specifyaconvenientpointinthemodulispacetostudy. WegivethescalarX avacuumexpectation
1
valueof v σ :
2 3
v
X = σ . (1.3)
1 3
h i 2
1TheLagrangedensitygivenin[6]isasupersetofthatforN =4SYMreproducedhere.[6]follows[7],butcorrects
severalmisprintswhichexistinthatreference.
2σ isthe2×2identitymatrix.
0
–2–
Giving the X a VEV in the prescribed manner has several consequences. The gauge group
1
SU(2)isspontaneouslybrokentoU(1)andtheSU(2)gaugemultipletof = 4isbrokenupinto
N
aU(1)gaugemultipletof = 4andamassivevectormultipletof = 4chargedundertheU(1).
N N
In the case of a spontaneously broken SU(2) symmetry, there is nothing special about the choice
we made for the VEV of X . This is because the symmetry breaking pattern SU(2) U(1)
1
→
is unique. For more complicated gauge groups, it is possible to obtain more than one symmetry
breaking pattern.
TheU(1)gauge multiplet iscomposed of theunbroken U(1) gauge field, A0, thefour Gold-
µ
stonespinorsofsuperconformal symmetry(allfoursuperconformal symmetriesarespontaneously
broken), ψ0, the five Goldstone bosons of R-symmetry (the R-symmetry is spontaneously broken
i
fromSO(6)withfifteengeneratorstoSO(5)withtengenerators[5]), X0,X0,Y0,Y0,Y0 ,and
{ 2 3 1 2 3}
the Goldstone boson of dilatations, X0. The massive vector multiplet is composed of a complex
1
± ± ± ± ± ± ±
vector boson, A , four Dirac spinors, ψ , and five complex scalars, X ,X ,Y ,Y ,Y .
µ i { 2 3 1 2 3 }
Each of these complex fields acquires a mass squared m2 = g2v2 after spontaneous symmetry
breaking.
Though it is not obvious, there is evidence that each of the noteworthy properties that the
= 4 S-matrix possessed in the conformal phase carries over to the Coulomb phase. That the
N
scatteringamplitudesarestillfreeofUVdivergencesfollowsfromtheargumentsin[13]. Whether
the scattering amplitudes are all expected to be pure scalar box is still somewhat speculative. It
was, however, shown fairly recently [23] that the complete low energy effective action of SU(2)
= 4SYMbroken toU(1)isconsistent withtheamplitudes being pure box. Thiswasdone via
N
an = 2 superspace calculation, where [23]allowed for background hyper-multiplets as wellas
N
background gaugemultiplets. Itshouldbestressedthattheresultsof[23]donotconstitute aproof
thattheCoulombphaseamplitudesarepurebox,sincetheeffectiveactionobtained in[23]isonly
validifallMandelstam invariants aresmallrelativetom.
In fact, to the author’s knowledge, the work of [23] is the first paper which attempts to treat
the complete SU(2) U(1) = 4 model; the other known treatments [24, 25] focus on the
→ N
masslesssectorofthespontaneously brokentheory. Oftheseworks,only[25]attemptstocompute
a scattering amplitude. They calculate the four X0 one-loop amplitude in the = 1 supergraph
2 N
formalism. Thesuperspace approach isnotwell-suited forcalculation offour-point functions, due
tothefactthatisnotstraightforwardtoseetheunderlyingsimplicityofresultsobtainedinthisway
(seee.g. [26]). Itshouldbestressed,however,thatthespiritoftheircalculationwasquiteprescient.
Theprimarymotivationof[25]wastostudy = 4supersymmetricscatteringamplitudeswiththe
N
IR divergences regulated in anatural way. Introducing amass-regulator viaspontaneous breaking
ofgaugesymmetryisactuallyoneofthemethodsbywhichAldayandMaldacena[27]regularized
theIRdivergences ofthefour-point gluonscattering amplitude in = 4atstrongcoupling.
N
Another interesting paper [28]rederives, inaslightly different context, thewell-known result
[20,21]thatthemasslesssector ofourmodelshould haveexactly thesamesupersymmetric Ward
identites between scattering amplitudes as conformal = 4. The authors of [28] point out that
N
useful supersymmetric Ward identities must still exist when some of the external scattering states
are massive. In fact, such Ward identities have already been applied to relate amplitudes with
massivequarks toknownamplitudes withmassivescalars[29].
Inthisworkwecalculate anumberoffour-point one-loop scattering amplitudes inthe
–3–
SU(2) U(1) = 4model. Weexamineboththecasewherealltheexternal legsaremassless
→ N
andthecasewheresomeofthemaremassive. Inparticular,weprovideevidencethatthescattering
amplitudes withmassiveexternal statesare,infact,purebox. Wefocusonhowthesymmetriesof
our model constrain theanswers obtained and, insomecases, allow ustorelate distinct scattering
amplitudestoeachother. Finallyweexplainhowourfourphotonone-loopscatteringamplitudein
theSU(2) U(1) = 4modelcanbethoughtofasamass-regulated, color-ordered, amplitude
→ N
in an unbroken SU(2) = 4 SYM theory. In other words, our work provides a weak-coupling
N
analogofthemassregulatorintroducedbyAldayandMaldacena[27]tocomputegluonscattering
amplitudes atstrongcoupling.
Theplanofthispaperisasfollows. InSectionTwo,wequantize theLagrangedensity
(Eqn. 1.1) in R gauge, motivate an efficient choice of gauge for the computations we want to
ξ
do, and address a subtlety in the Feynman rules. In Section Three, we present and discuss the
calculation of several one-loop four-point scattering amplitudes in both the massless and massive
sectors of = 4 SYM in the Coulomb phase. In Section Four, we summarize our results and
N
present some ideas for future work. In Appendix A, we provide the definitions of the master
integrals we use in Section Three. In Appendix B, we provide the complete Lagrange density of
ourmodelexpandedinaformwheretheFeynmanrulescaneasilybederived. Finally,inAppendix
C,wegivethediagrammaticexpansions ofallone-loop amplitudescalculated inSectionThree.
2. Setup
Weemploythemetricdiag( ,+,+,+)andintroduce thenotation
−
φ iφ
± 1 2
φ ± (2.1)
≡ √2
where φisageneric field. Inorder todo perturbative calculations inour model, wemustperform
R quantization on the classical Lagrange density given in the previous section with the VEV of
ξ
Eqn. 1.3. Thisisastandardcalculation andthedetailswillnotbeshownhere. Wenowgivethose
terms which were either affected by or introduced by the Fadeev-Popov procedure. These terms
includethosethatleadtothepropagatorsforthegaugefield,A0,themassivevectorfields,A±,the
µ µ
Goldstone fields, X±, the massless Higgs-like field X0, and the Fadeev-Popov ghosts c0,c± .
1 1 { }
Alsoincluded aretheinteractions betweentheghosts,Higgs,Goldstones, andgaugefields:
1 1 1
= A0 gµν∂2+∂µ∂ν 1 A0 A+ gµν∂2+∂µ∂ν 1 +m2gµν A−
LFP;2 − 2 µ − − ξ ν − µ − − ξ ν
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17) (cid:17)
1 1
X+ ∂2+ξm2 X− X0 ∂2 X0 c¯ ∂2+ξm2 c c¯0 ∂2 c0
− 1 − 1 − 2 1 − 1 − − − 2 −
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(2.2)
= ξmgX0c¯c+ξmgX−c¯c0 igc¯∂µ A0c +igc¯∂µ A−c0 +igc¯0∂µ A+c +h.c.
LFP;3 − 1 1 − µ µ µ
(2.3)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Theinteractionswiththeghostfieldsarenotrelevanttothecalculationsperformedinthispaper,but
–4–
areincluded forcompleteness. Thepropagators, ontheotherhand,areobviously quiteimportant.
i gµν lµlν(1−ξ) i gµν lµlν(1−ξ)
A0(l)A0(l) = − − l2 A+(l)A−(l) = − − l2+ξm2
h µ ν i (cid:16) l2 (cid:17) h µ ν i (cid:16) l2+m2 (cid:17)
i i
X+(l)X−(l) = − X0(l)X0(l) = −
h 1 1 i l2+ξm2 h 1 1 i l2
i i
c¯(l)c(l) = − c¯0(l)c0(l) = − (2.4)
h i l2+ξm2 h i l2
Traditionally, one-loop gaugetheorycomputations havebeenperformedin’tHooft-Feynman
gauge. It has been known, however, at least since the work of [3], that working in unitary gauge
(ξ ) has significant practical advantages in theories without UV divergences, like = 4.
→ ∞ N
The reason for this is easily understood; in = 4 on the Coulomb branch, one only has to
N
compute the box diagrams that arise and keep the pieces ofthem which arepure scalar box, since
everything else must cancel in the end3. If, for example, one wanted to compute the four Higgs
(denoted X0X0X0X0 hereafter) one-loop amplitude in ’t Hooft-Feynman gauge, there would be
1 1 1 1
eightindependent boxdiagramstoevaluate,whereasinunitary gauge,therewouldonlybethree4.
Inother words, thefactthatwehavetoperform relatively little integral reduction inthecom-
putation ofone-loop four-point functions in = 4,meansthathaving fewerdiagramstoevaluate
N
benefits us more than having Feynman rules which are renormalizable by power-counting. The
rest of the Lagrangian can be obtained in a straightforward (but tedious) way by making the shift
X X + X in Eqn. 1.1 and performing the traces over the SU(2) matrices. The many
1 1 1
→ h i
interaction terms obtained in this way are presented in Appendix B in a form where the Feynman
rulescanbereadoff.
Thereisonesomewhatcounter-intuitive resultthatarisesincarrying outthisprogram. Letus
consider the derivation of the propagators of the four Dirac fermions. Upon performing the shift
X X + X , we arrive at a fermion mass term ig tr ψ¯α1 [ X ,ψ ] from the Yukawa
1 → 1 h 1i − { i ij h 1i j }
interaction of the SU(2) Majorana fields with X . Expanding this out and adding in the Dirac
1
fields’kinetic termsgives
= Φ¯ ∂/Φ im(Φ¯ Φ Φ¯ Φ +Φ¯ Φ Φ¯ Φ ), (2.5)
Lψ¯ψ − i i− 1 2− 2 1 3 4− 4 3
wherewehaveintroduced thenotation
ψ¯+ Φ¯ ψ− Φ . (2.6)
i ≡ i i ≡ i
Wecannowdiagonalize themassmatrixinflavorspaceviatheunitarytransformation
Φ 1 i i Φ
1 = − 1 . (2.7)
Φ2! √2 1 1 ! Φ2!
e
3Thishasnotconclusivelybeenshownforallscatteringamplitudeswithmorethansixexternallegsintheconformal
phaseandispurelyspeculativeforamplitudesintheCoulombphasewhicehincludemassiveexternalstates.
4Technically,theghostloopwouldbenon-zerointhiscasealso,sincethecouplingofX0tothechargedghostscarries
1
an explicit factor of ξ. This, however, would not contributeascalar box integral but only amomentum independent
constant,whichwouldplayaroleinthecancellationoftheUVdivergencesifweexplicitlykepttrackofthem.
–5–
Aftermakingthischangeofvariables, thequadratic Lagrangedensity fortheDiracfieldsreads
′ = Φ¯ ∂/Φ m Φ¯ Φ Φ¯ Φ +Φ¯ Φ Φ¯ Φ , (2.8)
Lψ¯ψ − i i− 1 1− 2 2 3 3− 4 4
whichresultsinnon-standard propagators(cid:0)fortheΦ andΦ fields: (cid:1)
2 4
ie ie/l +m e e e e e e e ei i/l m
Φ¯ (l)Φ (l) = − − Φ¯ (l)Φ (l) = − − −
h 1 1 i l2+m2 e h 2e 2 i l2+m2
(cid:0) (cid:1) (cid:0) (cid:1)
i i/l +m i i/l m
Φ¯ (l)Φ (l) = − − Φ¯ (l)Φ (l) = − − − . (2.9)
h 3 3 i l2+m2 h 4 4 i l2+m2
e e (cid:0) (cid:1) e e (cid:0) (cid:1)
If we erroneously tried to give standard Dirac propagators to all four fields, we would be able to
find, amongest otheer things, one-loop four-scalar scatterieng ameplitudes whichviolate supersymme-
try. ThisΦbasisisaconvenient onetouseforperturbative calculations.
3. One-Loop Four-PointScattering Amplitudes
e
Before attempting to calculate an amplitude with massive external states, we first rederive a
knownresultinthemasslesssector[30]andtwoamplitudesrelatedtoitbyknownsupersymmetric
Wardidentites[19,28]. Theworkof[30]givesallindependentone-loophelicityamplitudesforthe
(SMorminimalSUSY)process 0 γγγγ. (Itisconvenient toconsider allparticles asoutgoing,
→
and weadopt this convention throughout.) This requires them to calculate the effect of a massive
scalarloop,amassivefermionloop,andamassivevectorloop. Thisisactuallymuchmorethanwe
needtodothecalculationin = 4. Thediagrammaticrepresentationsoftheamplitudesdiscussed
N
inthissectionaregiveninAppendixC.
First of all, as discussed above, we may neglect all terms in the results of [30] which do not
correspond to box integrals. Furthermore, two well-known supersymmetric Ward identities are
(k+,k+,k+,k+) = 0 and (k+,k+,k+,k−) = 0. It is easily seen by examining the proofs
M 1 2 3 4 M 1 2 3 4
of these relations [22] or the explicit calculations in [31] that these relations hold independent of
whether the particles running in the loops are massive. These SUSYrelations immediately tell us
that two of the independent one-loop A0A0A0A0 amplitudes are identically zero to all orders in
µ ν ρ σ
perturbation theory. Itfollows that (k+,k+,k−,k−)isthe only helicity amplitude which must
M 1 2 3 4
becalculated. Inaddinguptheboxcoefficientsgivenin[30],itisimportanttorememberthatthere
arefivescalarloops,fourfermionloops,andonevectorloop5. Addingupthepiecesgivesthefinal
result
(k+,k+,k−,k−) = 8g4s2 I(4)(s,t)+I(4)(s,u)+I(4)(t,u) , (3.1)
M 1 2 3 4 0 0 0
(cid:16) (cid:17)
whereweadopttheconventions
s= (k +k )2 t = (k +k )2 u= (k +k )2 (3.2)
1 2 1 4 1 3
(4) (4) (4)
and the scalar box integrals I (s,t), I (s,u), and I (t,u) are defined in Appendix A. To
0 0 0
calculate the helicity amplitude weused the standard non-covariant basis forthe polarization vec-
tors, where one works in the center-of-mass frame and expresses all non-zero dot products of the
polarization vectorswitheachotherandtheexternal momentaintermsofs,t,andu.
5SeeAppendixCfordetails.
–6–
ToreproducetheresultofEqn. 3.1,weworkedinunitarygauge,asexplainedinSectionTwo,
andemployed theMathematica package FeynCalc6 [32]. Aninteresting featureofthecalculation
isthattheloopmomentumpolynomialsexplicitlycancelamongstthevariouscomponents. Inother
words, forthis particular calculation, the integral reduction ofthe box graphs istrivial if one adds
upthecomponents first.
To illustrate this point, we give the loop momentum numerators of the (properly weighted)
(4)
components whichleadtothecoefficientofI (s,t):
0
5 N scalar(k+,k+,k−,k−) = 160l ǫ+(k )l ǫ+(k )l ǫ−(k )l ǫ−(k ) (3.3)
× M 1 2 3 4 · 1 · 2 · 3 · 4
(cid:18) (cid:19)
4 N fermion(k+,k+,k−,k−) = 32sl ǫ+(k )l ǫ+(k )+32sl ǫ−(k )l ǫ−(k )
× M 1 2 3 4 · 1 · 2 · 3 · 4
(cid:18) (cid:19)
256l ǫ+(k )l ǫ+(k )l ǫ−(k )l ǫ−(k ) (3.4)
1 2 3 4
− · · · ·
N vector(k+,k+,k−,k−) = 8s2+96l ǫ+(k )l ǫ+(k )l ǫ−(k )l ǫ−(k )
M 1 2 3 4 · 1 · 2 · 3 · 4
(cid:18) (cid:19)
32sl ǫ+(k )l ǫ+(k ) 32sl ǫ−(k )l ǫ−(k ). (3.5)
1 2 3 4
− · · − · ·
Adding up Eqns. 3.3-3.5 gives 8s2, which immediately lets us read off the I(4)(s,t) contribution
0
to Eqn. 13 above. This additional cancellation structure between the numerator loop momentum
polynomials of diagrams with different internal structure is related to the fact that the external
states are gauge fields. This can be understood by thinking about the calculation in background
fieldgauge[3].
In fact, the same cancellation structure observed above is present also in the case of the
A0A0X0X0 amplitude7. Note that, in this case, there is a supersymmetric Ward identity which
µ ν 2 2
sets the amplitudes with identical helicities to zero. For the non-zero helicity configurations, we
find
(k+,k−) = (k−,k+) = 8g4tu I(4)(s,t)+I(4)(s,u)+I(4)(t,u) . (3.6)
M 1 2 M 1 2 0 0 0
(cid:16) (cid:17)
An interesting point is the following. Suppose we took the result for the A0A0A0A0 ampli-
µ ν ρ σ
tude and tried to use supersymmetric Ward identities to predict the answer for the A0A0X0X0
µ ν 2 2
amplitude. The prediction of the supersymmetric Ward identities in the conformal phase must be
identical to that considered here, since the identities are independent of color and the difference
between the two calculations shows up only in the basis of integrals, not in the coefficients. In
otherwords,itshouldbepossibletoshowusingonlyN = 4supersymmetry that
s2
(k+,k+,k−,k−)= (k+,k−) (3.7)
M 1 2 3 4 tuM 1 2
It is clear from the = 4 supersymmetry that there must be some proportionality between
N
thetwoamplitudes. Tofindtheproportionality constant,itisenoughtoconsidertheusualleading-
colortreeamplitudesinunbroken = 4,duetothecolor-independentandnon-perturbativenature
N
6FeynCalcisaframeworkforperformingperturbativecalculationsingaugetheories.
7ItshouldbestressedthatwechooseX0forconcreteness. Aswillbediscussedbelow,theN = 4supersymmetry
2
ofthemodeldemandsthat,alternatively,wecouldhavereplacedX0withanyof{X0,X0,Y0,Y0,Y0}andarrivedat
2 1 3 1 2 3
theresultofEqn.3.6.
–7–
ofthesupersymmetricWardidentites. Theleading-colortreeamplitudeforfourgluonscatteringis
3,4 4
(k+,k+,k−,k−) = h i (3.8)
A 1 2 3 4 1,2 2,3 3,4 4,1
h ih ih ih i
andtheleading-color treeamplitudefortwogluontwoscalarscattering is
2,3 2 2,4 2
(k+,k−)= h i h i . (3.9)
A 1 2 1,2 2,3 3,4 4,1
h ih ih ih i
Theratioofthesetwotreesis
(k+,k+,k−,k−) 3,4 4 s2
A 1 2 3 4 = h i = C , (3.10)
(k+,k−) 2,3 2 2,4 2 tu
A 1 2 h i h i
whereCisanunimportant overallphase.
Thisanalysisillustratesthatanyotherfour-pointamplitudeinthemasslesssectorofourmodel
must have mass-independent coefficients, since exactly the same structures would appear if we
were in the conformal phase of the theory and we have already seen that the loop integral coeffi-
cients of (k+,k+,k−,k−) and (k+,k−) have no dependence on m. This is a direct conse-
M 1 2 3 4 M 1 2
quenceofthefactthatthesupersymmetric Wardidentities arethesameinbothcases,independent
ofwhethertheparticles running intheloopsacquiremass.
Before leaving the massless sector, we calculate the one-loop X0X0X0X0 amplitude and
2 2 2 2
find a result that disagrees with that of [25]. As before, an analysis based on the supersymmetric
Ward identities predicts a unique answer for the four scalar amplitude as a rational function of
s, t, and u times the four photon amplitude. In other words, supersymmetry tells us we could
equally well choose to compute the amplitude with four of any of the other massless scalars,
X0,X0,Y0,Y0,Y0 , since, in the supersymmetric Ward identity argument, the flavor of the
{ 1 3 1 2 3}
scalarbeingscattered wasirrelevant.
Wenowaddress apointwhichwasglossedoverinthepreviouscalculation. Since
X0,X0,Y0,Y0,Y0 transform as a 5 of the manifest SO(5) R-symmetry, the X0X0X0X0,
{ 2 3 1 2 3} 2 2 2 2
X0X0X0X0,oranyoftheY0Y0Y0Y0 amplitudeslookexactlythesamegauge-invariant compo-
3 3 3 3 q q q q
nent by gauge-invariant component8. What we mean is that the sum of the graphs with fermionic
e
loops will be the same for each field in the R-symmetry multiplet and the sum of the graphs with
bosonic loopswillbethesameforeachfieldintheR-symmetrymultiplet.
On the other hand, it is not entirely clear from the outset how the X0X0X0X0 amplitude
1 1 1 1
worksouttobeidenticalwiththeothermasslessscalarfour-pointamplitudes. Thedifficultyisthat
X0 is a singlet of SO(5) and its interactions are rather different than those of the other massless
1
scalars. To clarify this, wegive the gauge-invariant components for both situations and show that
theyleadtothesameresult. Wefind
Boseloops = 8g4(s2+t2)I(4)(s,t)+8g4(s2+u2)I(4)(s,u)
M5 0 0
+8g4(t2+u2)I(4)(t,u) (3.11)
0
Feermiloops = 8g4stI(4)(s,t)+8g4suI(4)(s,u)+8g4tuI(4)(t,u) (3.12)
M5 0 0 0
8Itis,however,instructivetocomparethefermionicloopgraphswithexternalscalarstothosewithpseudo-scalars
usingtheLagrangedensitygiveninAppendixB.
e
–8–
forthecaseoftheX0X0X0X0, X0X0X0X0,oranyoftheY0Y0Y0Y0 amplitudes and
2 2 2 2 3 3 3 3 q q q q
Boseloops = 8g4(s2+t2+32m4)I(4)(s,t)+8g4(s2+u2+32m4)I(4)(s,u)
M1 0 0
+8g4(t2+u2+32m4)I(4)(t,u) (3.13)
0
Feermiloops = 8g4(st 32m4)I(4)(s,t)+8g4(su 32m4)I(4)(s,u)
M1 − 0 − 0
+8g4(tu 32m4)I(4)(t,u) (3.14)
− 0
e
for the X0X0X0X0 amplitude. In either case, summing the contributions and using the kine-
1 1 1 1
matic relation s+t+u = 0 gives Eqn. 3.15 below. Although the individual components of the
X0X0X0X0 amplitude calculation hadnon-trivial massdependence, thisdependence cancels out
1 1 1 1
in the finalresult, as weargued itmust for allamplitudes in the massless sector of the theory. We
findthatEqn. 4.9of[25]shouldread
4×X20 = 4g4(s2+t2+u2) I(4)(s,t)+I(4)(s,u)+I(4)(t,u) . (3.15)
M 0 0 0
(cid:16) (cid:17)
Finally, we compute an example of a one-loop scattering amplitude with massive external
states. For simplicity, we study the X0X0X+X− amplitude. It took somewhat more effort to
2 2 2 2
check thisamplitude sinceallprevious workhasfocused exclusively onthemassless sector ofthe
theory. We performed all calculations both in unitary and ’t Hooft-Feynman gauge. This check,
however, is not sensitive to problems originating from the fermionic box graphs. It was useful to
also keep track ofalltriangle diagrams andexplicitly show thatnoscalar triangle integrals appear
inthefinalresult. TheFeynArts[33,34]package(asanadd-onforFeynCalc)wasusedtogenerate
theFeynmandiagrams. Wefind
X20X20X2+X2− = 2g4(s2+(t+m2)2+(u+m2)2) I(3) (s,t)+I(3) (s,u) , (3.16)
M 2hard 2hard
(cid:16) (cid:17)
(3) (3)
where,asbefore, thescalarboxintegralsI (s,t)andI (s,u)aredefinedinAppendixA.
2hard 2hard
It is remarkable that, apart from a factor of two, the X0X0X0X0 and the X0X0X+X−
2 2 2 2 2 2 2 2
amplitudes bothhaveexactlythesamecoefficient structure. Thiscanbeeasilyseenbycomparing
thekinematics ofthetwocases. ForX0X0X0X0 wehave
2 2 2 2
2k k = s 2k k = u 2k k = t (3.17)
3 4 1 3 1 4
· · ·
andforX0X0X+X−,provided wechoosek2 = m2,
2 2 2 2 1 −
k2 = m2,k2 = 0, andk2 = 0,wehave
2 − 3 4
2k k = s 2k k = u+m2 2k k = t+m2. (3.18)
3 4 1 3 1 4
· · ·
Upon examining the form of Eqns. 3.15 and 3.16, we see that the scattering amplitudes for
these two processes are very similar. The fact that Eqns. 3.15 and 3.16 are so similar suggests
that the technical complications arising from the inclusion of massive external states in = 4
N
scattering amplitudesarenotaseriousissueandthatasystematicanalysisofourmodel’sS-matrix
should bestraightforward tocarryout.
–9–