WW Scattering in Walking Technicolor

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by  Roshan Foadi

WW Scattering in Walking Technicolor
Roshan Foadi∗ and Francesco Sannino†
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.
We analyze the WW scattering in scenarios of dynamical electroweak symmetry breaking of
walking technicolor type. Weshow that in these theories there are regions of theparameters space
allowed bytheelectroweak precision data, inwhichunitarityviolation isdelayedat treelevelupto
around 3-4 TeV without the inclusion of any sub-TeV resonances.
The simplest argument often used to predict the ex- tors, not gauged under the technicolor gauge group, are
istence of yet undiscovered particles at the TeV scale calculable to any order in perturbation theory. A rel-
comes from unitarity of longitudinal gauge boson scat- evant question to ask is whether a walking regime can
8
teringamplitudes. Iftheelectroweaksymmetrybreaking be achievedwitha sufficientlysmallnumberoffermions.
0
0 sector (EWSB) is weakly interacting, unitarity implies In the context of SU(NTC) gauge theories with fermions
2 that new particle states must show up below 1 TeV, be- in the fundamental representation, a large number NTF
ing these spin-0 isosinglets (the Higgs boson) or spin-1 of techniflavors is required to achieve walking dynam-
n
a isotriplets (e.g. Kaluza-Klein modes). A strongly in- ics, even for small values of NTC. If all technifermions
J teracting EWSB sector can howeverchange this picture, have electroweak charge,this results in a large contribu-

4 because of the strong coupling between the pions (eaten tion to S and other electroweak parameters. Recently,
by the longitudinal components of the standard model however, a new class of SU(NTC) gauge theories with
]
gaugebosons)andtheotherboundstatesofthestrongly fermionsinhigher-dimensionalrepresentationshavebeen
h
p interacting sector. An illuminating example comes from arguedto displaynear-to-conformalbehavioralreadyfor
- QCD. In Ref. [1] it was shown that for six colors or smallvaluesofNTF andNTC[19]. Acompletecatalogue,
p
more,the 770 GeV ρ meson is enoughto delay the onset basedontheSchwinger-Dysonapproximation,ofallpos-
e
h of unitarity violation of the pion-pion scattering ampli- sible SU(NTC) walking technicolor gauge theories with
[ tude upto wellbeyond1GeV. Here the ’t HooftlargeN fermions in a given but arbitrary representation can be

limit was used, however an even lower number of colors found in [20]. An all order β-function for any nonsuper-
1
v is needed to reach a similar delay of unitarity violation symmetric, strongly interacting, and asymptotically free
3 when an alternative large N limit is used [2]. Scaling gaugetheoryhasbeensuggestedin[21],therebyallowing
6 up to the electroweak scale, this translates in a 1.5 TeV newmethodsforthe investigationofnonsupersymmetric
6
techni-ρ being able to delay unitarity violation of longi- conformalwindows. Thefirsthintsofwalkingorpossibly
0
tudinal gauge boson scattering amplitudes up to 4 TeV conformal dynamics for gauge theories with fermions in
.
1 or more. Such a particle would be harder to be discov- higher dimensional representations has appeared in [22],
0
eredattheLHCandILC.Suchamodel,however,would and a first benchmark for a working model of WT has
8
not be realistic for other reasons: a large contribution been constructed in [16]. Finally it should be mention
0
: to the S parameter[3], andlargeflavorchangingneutral that in this framework unification of the SM gauge cou-
v
currents (FCNC) if the ordinary fermions acquire mass plings can be achieved [14].
i
X via an old fashioned extended technicolor sector (ETC), An alternative scenario in which the S parameter is
r to mention the most relevant ones. naturally suppressed is provided by Custodial Techni-
a
Walking Technicolor (WT) [4, 5, 6] provides a natu- color (CT) [15], in which an enhanced global symme-
ral framework to address these problems. In fact walk- try [23] prohibits interactions between the spin-0 and
ing dynamics helps suppressing FCNC without prevent- the spin-1 sectors of the theory, thereby protecting all
ing ETC from yielding realistic fermion masses. Notice of the elctroweak parameters against large corrections.
that it is always possible to resort to new scalars to This feauture, however, renders the vector mesons to-
give mass to the ordinary fermions while having tech- tally helpless when it comes to unitarize the pion-pion
nicolor only in the gauge sector, or even marry techni- scatteryamplitudes. Therefore,aHiggsbosonwithmass
color with supersymmetry [7, 8, 9, 10, 11, 12, 13, 14]. aroundorbelow1TeVisalwaysneededinCT.Basedon
Furthermore certain WT models are in agreement with these premises, it is interesting to analyze the pion-pion
the constraints imposed by electroweak precision data scattering in generic models of WT.
(EWPD) [15, 16, 17], since the walking dynamics it- Inordertoextractpredictionsinpresenceofastrongly
self naturally lowers the contribution to the S param- interacting sector and an asymptotically free gauge the-
eter relative to a running theory [18]. Besides, new lep- ory, we make use of the time-honored Weinberg sum
tonicsectors[17],whichmaybe neededtoavoidpossible rules(WSR)[24],whicharestatementsaboutthevector-
Witten topological anomalies, can render the overall S vector minus axial-axial vacuum polarization functions,
parameter negative. The contributions from these sec- known to be sensitive to chiral symmetry breaking. As-
2
suming a low energy spectrum consisting of a narrow 6
vector-vector resonance and a narrow axial-vector reso-
5
nance, the first WSR reads
2 2 2 VL4
FV−FA =Fπ , (1) Te 3
H
V
whereFV (FA)isthedecayconstantofthevectorial(ax- M 2
ial) resonance, and Fπ is the pion decay constant. In
technicolor models Fπ = 1/√2GF = 246 GeV. The sec- 1
ondWSR,unlikethefirstone,receivesimportantcontri- 0
butions from throughout the near conformal region, and 0 1 2 3 4 5 6
reads [16, 18] M TeV
AH L
8π2
F2M2 F2M2 =a F4 , (2) FIG. 1: MV as a function of MA for S = 0.1, and different
V V A A π
− d(R) valuesofg. Thecurvewiththelowestslopecorrespondstothe
lowest valueof g compatible with therequirement ΓV/MV
≤
where d(R) is the dimension of the representation of the 0.5. The curve with the largest slope corresponds to g =
underlyingfermions,andaisexpectedtobepositiveand p8π/S: valuesof g abovethisboundgiveFA2 <0, andmust
(1). In case of running dynamics we obtain a=0, and therefore berejected. Seebelow for theallowed region inthe
Othe secondWSRrecoversits familiarform. The parame- (MA,g) plane.
teraisanon-universalquantitydependingonthedetails
of the underlying gauge theory expected to be positive
andO(1). Anyotherapproachtryingtomodelthewalk-
ing behavior will have to reduce to ours. The fact that
a is positive and of order one in walking dynamics it is
supported, indirectly, alsofrom the workof Kurachiand
Shrock[25]. Attheonsetofconformaldynamicstheaxial
and the vector will be degenerate, i.e. MA = MV = M,
usingthefirstsumruleonefindsviathesecondsumrule Recent computations further support the reduction of S
a = d(R)M2/(8π2Fπ2) leading to a numerical value of in walking theories [26].
about 4- 5 from the approximate results in [25]. We will
howeveruse only the constraintscoming from the gener-
alizedWSRsexpectingthemtobelessmodeldependent.
Asaphenomelogicallyinterestingexampleweconsider
Thereisalsoa“zeroth”sumrule,whichisnothingbut
minimal walking technicolor (MWT) [16, 17, 19]. This
the definition of the S parameter:
is an SU(2) gauge theory with two fermions in the ad-
F2 F2 joint representation. If these form an electroweak dou-
V A
S =4π . (3)
(cid:20)M2 − M2(cid:21) blet, the perturbative contribution to the S parameter
V A
will be S = d(R)/6π = 1/2π 0.16. MWT is arguably
≃
Using Eq. (1) and Eq. (2) this becomes the WT model with the lowestS. This value canbe fur-
therreducedbyaddingtothe theorynewleptons(which
2 1 1 8π2Fπ2 inMWT arerequiredtocurethe Wittenanomaly),with
S =4πF + a . (4)
π(cid:20)M2 M2 − d(R)M2M2(cid:21) weak SU(2) mass splitting. With these ingredients the
V A A V
full estimatesfor S andT wereshownto be within 1σ of
The first two terms of this equation are the ordinary
theexperimentalexpectationvaluesforawideportionof
QCD-like contributions. The third term is negative and
theparameterspace[16,17,19]. Thisistruealsoforthe
ofthesameorderofthefirsttwo. Therefore,iftheρand
morestringesttestsbasedonthepreciseLEPparameters
a1 mesons of QCD were resonances of a walking theory, of Barbieri et. al. [27] explored for WT and CT in [15].
the corresponding S parameter would be considerably
TakingS (0.1,0.3)asarealisticestimateinMWT,and
lower than the QCD value S 0.3. As mentioned be- ∈
≃ using Eq. (1) and Eq. (3), allows us to take FV and MA
fore, this shows that the walking dynamics is not only
(for example) as the only independent inputs, since Fπ
importantinsuppressingFCNC,butalsoinloweringthe
is known. In alternative to FV we can take the “gauge
contributiontotheS parameterrelativetoatheorywith
coupling”g ofaneffectivemodel[16]inwhichthevector
a running coupling constant. Since QCD’s S is approxi-
mesons are treated as gauge fields. FV is given by
matelytwice itsperturbativeestimate,itappearssafe to
estimate S for any WT theory to be within 100% the 2M2
2 V
± F = . (5)
perturbative value, calculated from techniquark loops. V g2
3
4
0.4
3.5
0.2 3
VL2.5
a00 0 Te 2
H
-0.2 (cid:143)!!!s 1.5
1
-0.4 0.5
0 1 2 3 4 0 2.5 5 7.5 10 12.5 15
(cid:143)!s!! TeV g
H L
(a) (b)
FIG. 2: (a) I=0, J=0 partial wave amplitude for MA=1.5 TeV, S = 0.1, and three different values of g. The central curve
corresponds to the largest delay of unitarity violation, up to √s 3.7 TeV. The lowest curve corresponds to a slightly lower
≃
valueofg,andviolatesunitarityjustbelow3.7TeV.Theuppercurvecorrespondstoaslightlylargervalueofg,butitviolates
unitarity at a much lower energy, √s 1.9 TeV. As a consequence there is a discontinuity in the plot of unitarity violation
≃
energy as a function of g (b).
From Eq. (1), Eq. (3), and Eq. (5) we find Wecanusetheseresultstoanalyzetheπ πscattering.
−
WeusenonlinearrealizationsinwhichtheHiggsbosonis
2 2 2
2 g S 2 g Fπ integratedout. Forthecorrespondingtree-levelinvariant
M = 1 M + (6)
V A
(cid:18) − 8π (cid:19) 2 amplitudes and scattering formalism see Appendix A of
g2S 2M2 Ref. [28] or directly equations (1) and (2) in [2] after
2 A 2
F = 1 +F (7)
V (cid:18) − 8π (cid:19) g2 π havingsetto zerothe massofthe pions. InFig.2(a)we
g2S 2M2 plot the I =0, J =0 partial wave, a00, for MA=1.5 TeV
F2 = 1 A . (8) and three different values of g. The central curve has
A (cid:18) − 8π (cid:19) g2 an a0 = 0.5 maximum, and displays the largest value of
0
unitarity violation energy, around 3.7 TeV, for MA=1.5
Eq. (8) immediately gives an upper bound for g:
TeV.Thelowestcurvehasaslightlylowervalueofg,and
violatesunitarityjustbelow3.7TeV.Howevertheupper
8π
g < . (9) curve violates unitarity at a much lower energy, around
r S
1.9 TeV. This shows that there is a discontinuity in the
This is represented by the upper straight line of Fig. 3 plot of unitarity violation energy as a function of g, as
(a), for S = 0.1 (top), S = 0.2 (center), and S = 0.3 shown in Fig. 2 (b). The location of this discontinuity
(bottom). Additional upper and lower bounds are given correspondsthereforetoa“critical”valueofg,gc. While
by the requirement that the vector mesons are narrow above gc the amplitude is not sufficiently unitarized by
states,sincethisisanecessaryconditionfortheWSR’sto the vector-vector meson, and a spin-0 isospin-0 state is
hold. Above the upper left solidcurveMV is largerthan required with a pole mass near the energy scale where
2MA,andthedominantV A+Adecaychannelopens unitarity is violated [1], below and near gc the theory
→
up,withalargecontributiontoΓV. Belowthelowerthin may very well be Higgsless. Moreover, since MA < MV
solid curve ΓV(ππ)/MV is larger than 0.5. Here ΓV(ππ) for MA =1.5 TeV and S = 0.1 (see Fig. 1), these states
is the partial width for decays into two π’s, which turns are more difficult to be discovered at the LHC or ILC
out to be reach. Of course this argument does not exclude the
presence of light states, but shows clearly that a worst
g2 M5
ΓV(ππ)= 9V6πππMV = 96πgV2Fπ4 . (10) ceaxscelusdceednaarsioitoisfgneonedreatlelycteioxnpeocftendewinpmaordtiecllseswictahnanowtebake
EWSB sector.
Where in the last equation we used the relation gVππ =
M2/(gF2). A final constraint is the requirement that Notice that for small values of g the theory seems to
V π
a is a positive number and (1). On the lower thick lose unitarity rather soon, even below the standard 1.2
O
curve a = 0, and both WSR’s are satisfied in a running TeVbound ofthe chiralLagrangian. This seeminglyun-
regime. Below the curve a is negative. Above the curve reasonableresultcomesfromholdingMA fixed,inwhich
a is positive and (1), and all corresponding values of case taking small values of g corresponds to taking large
O
MA and g are possible. values of gVππ. In this limit, and for relatively small
values of MV, the interaction between the vector-vector
4
20 10
17.5
8
15
12.5 VL 6
e
g 10 T
H
7.5 (cid:143)!!!s 4
5
2
2.5
0 1 2 3 4 5 6 0 1 2 3 4 5 6
M TeV M TeV
AH L AH L
20 10
17.5
8
15
12.5 VL 6
e
g 10 T
H
7.5 (cid:143)!!!s 4
5
2
2.5
0 1 2 3 4 5 6 0 1 2 3 4 5 6
M TeV M TeV
AH L AH L
20 10
17.5
8
15
12.5 VL 6
e
g 10 T
H
7.5 (cid:143)!!!s 4
5
2
2.5
0 1 2 3 4 5 6 0 1 2 3 4 5 6
M TeV M TeV
AH L AH L
(a) (b)
FIG. 3: From top to bottom these figures correspond to S = 0.1, S = 0.2, and S = 0.3. (a) The dashed line corresponds to
gc as a function of MA. The thin solid lines give bounds on g and MA coming from self consistency (no imaginary numbers)
and the requirement that the vector-vectormeson is a narrow state, as explained in thetext. The thick solid line corresponds
to a=0: along this curve both WSR’s are satisfied in a running regime, while below the curve a<0, and the corresponding
values of g and MA must be rejected. This curve is hit by gc, with a narrow width, for S =0.3, but not for S 0.2, proving
≤
that QCD-like theories are only unitarized, at the tree-level, by vector mesons for large values of S or, which is the same, a
largenumberofcolors. (b)Unitarityviolationscalealongg=gc. TotherightoftheverticalthinlineΓV/MV becomesgreater
than 0.5. To theright of the thick verical line a becomes negative.
meson and the pions becomes too strong,and the model violation is delayed to higher energy scales, and a spin-
quickly looses unitarity. Another way to see this comes 0 isospin-0 state is not needed with mass below 1 TeV.
from Eq. (5): taking the g 0 limit with MV fixed Notice that for S = 0.1 (top) and S = 0.2 (center) gc
→
gives FV . It would be more physical to let MV is only attained in the running regime, i.e. a = 0, when
→ ∞
rapidly approach zero at the same time, in such a way ΓV/MV > 0.5, while for S = 0.3 one still has ΓV/MV <
that FV 0 as well. However here we only focus on 0.5. This is in agreement with the results of Ref. [1, 2],
→
the phenomenologically relevant regions of the parame- and shows that QCD-like theories are unitarized by the
ter space,in which the vector mesons are expected to be vector meson only for large values of S. In Fig. 3 (b)
at least in the 102 GeV range. the unitarity violation energy for g = gc is plotted as a
function of MA. To the right of the vertical thin line
In Fig. 3 (a) the dashed curve gives gc as a function ΓV/MV becomes greater than 0.5. To the right of the
of MA. Below and in the vicinity of this curve unitarity
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