Table Of ContentPion masses in quasiconformal gauge field theories
Dennis D. Dietrich and Matti J¨arvinen
Center for High Energy Physics, University of Southern Denmark, Odense, Denmark
(Dated: January 22, 2009)
WestudymodificationstoWeinberg-likesumrulesinquasiconformalgaugefieldtheories. Beyond
the two Weinberg sum rules and the oblique S parameter we study the pion mass and the X
parameter. Especially, we evaluate the pion mass for walking technicolour theories, in particular
alsominimalwalkingtechnicolour,andfindcontributionsoftheorderofuptoseveralhundredGeV.
PACSnumbers: 11.15.-q,12.15.-y,12.60.Nz
Introduction. The Large Hadron Collider (LHC) has tribution from the electroweak sector is phenomenolog-
beenbuilttohelpclarifyingthemechanismforthebreak- ically advantageous. Below we will see that quasicon-
9 ing of the electroweak symmetry. Beyond the standard formal dynamics lead indeed to an enhancement of the
0
model, where the symmetry is broken by an elementary aformentioned magnitude.
0
scalar Higgs particle, technicolour (TC) [1] provides a
2
possible mechanism which overcomes some of the defi-
n
ciencies of the former. In TC the electroweak symme- Pion mass. The Weinberg sum rules are given by [7],
a
try is broken by chiral symmetry breaking among addi-
J (cid:90) ds (cid:90)
tional fermions charged under the electroweak and the
2 Im Π(s)=0 and ds Im Π(s)=0, (1)
TC gauge groups. Three of the possibly more numer- s
2
ous Nambu–Goldstone modes are absorbed as longitudi-
respectively, where
] nal degrees of freedom of the electroweak gauge bosons.
h
Walking [2], that is, quasiconformal TC theories with (cid:18) q q (cid:19)
-p techniquarks in higher dimensional representations are Πaµbν(q)= gµν − µq2ν δabΠ(q2), (2)
p compatible with currently available precision data [3, 4].
e Likeeverywhereinstronglyinteractinggaugetheoriesef- and
h
fectiveLagrangiansareheavilyused[5,6]. Inmanycases
[ (cid:90)
the predictive power of these approaches is enhanced by Πab =−i d4xe−iq·x(cid:104)Va(x)Vb(0)−Aa(x)Ab(0)(cid:105). (3)
µν µ ν µ ν
1 enforcing the Weinberg sum rules [5, 6, 7].
v
8 AsymptoticallyfreegaugefieldtheoriesobeytheWein- VµaandAaµstandforthevectorandaxial-vectorcurrents,
berg sum rules [7]. It has been shown, however, that the
2 respectively,withflavourindexa∈{1,...,N2−1}. The
secondsumruleismodified[8]inquasiconformaltheories f
5 oblique S parameter [9] reads,
3 while the expression for the first remains unchanged. In
1. uthnechsaanmgeedse.nsOenaclesoththeemoobdliiqfiueedSsepcaornadmseutemr [r9u]leremisaiimns- S =(cid:90) ds Im Π¯(s), (4)
0 4π s2
9 posed together with the unmodified first, though, the S
0 parameterisreducedinquasiconformaltheories[8]. Here where Π¯ is Π without the contribution from Goldstone
: westudyrelatedquantities, concretely, theX parameter bosons. The X parameter [12] is obtained from
v
[12]andtheelectroweakcontributionstothepionmasses
i
X [10, 11]. The bare expression for the X parameter does X =(cid:90) ds Im Π¯(s). (5)
r not receive any corrections, but its value is reduced after 4πf2 s3
a π
the modified second sum rule is enforced simultaneously.
To the contrary, the pion mass is directly modified in The pion mass matrix falls into two factors [11]
quasiconformal theories.
m2 =mˆ2 ×m2, (6)
In the context of TC said mass is that of the Nambu– ab ab
Goldstone modes which are not absorbed as longitudi- wheremˆ2 dependsontheembeddingoftheelectroweak
ab
nal degrees of freedom of the electroweak gauge bosons. gauge group in the flavour symmetry group and m2
Firstofall,thesignofthesquaredmassdecideswhether stands for the overall magnitude [10],
the embedding of the electroweak gauge group in the
flavour symmetry group is stabilised or destabilised by 3(cid:0)g2+g2(cid:1)(cid:90) ∞
m2 =− 1 2 dslogs ImΠ¯(s), (7)
electroweak radiative corrections [11]. In the latter case 64π2f2
π 0
an additional mechanism is needed to stabilise the the-
ory and make it complete. Secondly, to date, no degrees where g and g are the weak coupling constants and
1 2
of freedom which could correspond to these modes have f is the pion decay constant. For the last relation the
π
been detected. Therefore, they have to be sufficiently result for the second Weinberg sum rules has been used
massivetohaveelludeddetection. Hence,asizeablecon- and is important to guarantee the convergence of the
2
integral. The overall magnitude m2 will be influenced gluons governs the form of the spectral function. Let us
by the quasiconformal dynamics. denote this part as ∆(s). Contributions beyond Λ2 are
The chiral symmetry breaking is to proceed from the negligible, such that
unbrokenflavoursymmetrygroupGtotheresidualgroup
H. The generators spanning H are to be called Sa while Im Πwalk =Im Πrun(s)+∆(s). (15)
the rest of the generators of G be called Xa. Following
Ithastobenoted,thatthethefirstaddendaboveissup-
the definition of [11] we normalize the generators Λi to
posed to have a structure like in the running case, but
which the electroweak fields couple as
need not be identical down to the values of the parame-
L=(cid:88)ψ¯γµΛiψAi, (8) ters. It turns out that in the first sum rule,
µ
i (cid:90) ds
∆(s)≈0 (16)
whereψ isthetechniquarkfield,Λi isamatrixinflavour s
space and the sum is over the electroweak gauge fields.
becausetheintegralistooconcentratedaroundtheorigin
Notice that the Λi also include the corresponding cou-
inordertobesensitivetothemodification. Forthesame
pling constants. They are linear combinations of both
reason, the expression for the S and X parameters are
categoriesofgenerators,Λi =Λi +Λi ,whereΛi stands
S X S even less affected. To the contrary [8],
for the contribution from unbroken generators and Λi
X
for the contribution from broken generators. Then (cid:90)
ds ∆(s)=−8π2αf4/d , (17)
π R
mˆ2 = 8 (cid:88)Tr(cid:8)(cid:2)Λi,(cid:2)Λi,Xa(cid:3)(cid:3)Xb −
ab g2+g2 S S where d is the dimension of the representation of the
1 2 i R
−(cid:2)Λi ,(cid:2)Λi ,Xa(cid:3)(cid:3)Xb(cid:9), (9) gauge group under which the fermions transform. α is
X X expected to be positive and of order one. The contribu-
tion to m2 is given by,
where we used the normalisation TrXaXb = δab/2 =
TrSaSb. 3(cid:0)g2+g2(cid:1)(cid:20) λ2 λ2 (cid:21)
For a running theory and assuming saturation by the m2 = 1 2 F2 ln −F2ln . (18)
walk 64π2f2 V M2 A M2
lightest resonances, we have π V A
Im Π (s) = Im Π¯ (s)−f2sδ(s), The scale λ is defined through the condition
run run π
Im Π¯run(s) = FV2δ(s−MV2)−FA2δ(s−MA2), (10) (cid:90) Λ2 λ2
dsln ∆(s)=0. (19)
s
where F and F are the vector and axial-vector de- 4Σ2(0)
V A
cay constants, respectively. Multiple resonances can be
Hence, λ ∈ [2Σ(0),Λ] and it can be estimated to lie ar-
included straightforwardly by summing over them. For
round the geometric mean of 2Σ(0) and Λ.
this spectral function we obtain from the sum rules,
Next, we use the first and the modified second Wein-
F2 F2 berg sum rule to eliminate the decay constants from the
MV2 − MA2 =fπ2 and FV2 −FA2 =0, (11) expressions for S, X and the pion mass:
V A
S M2 +M2 8π2f4 1
for the oblique parameters =f2 V A −α π ,
4π π M2M2 d M2M2
V A R V A
F2 F2 S
V − A = , (12)
M4 M4 4π
V A X M4 +M2M2 +M4 8π2f4M2 +M2
MFV26 − MFA26 = 4πXf2, (13) 4πfπ2 =fπ2 V MVV4MAA4 A −α dRπ MVV4MA4A,
V A π
and for the common factor of the pion masses, 3(cid:0)g2+g2(cid:1)(cid:20) M2M2 M2
m2 = 1 2 V A ln A +
3(cid:0)g2+g2(cid:1)F2 M2 walk 64π2 MA2 −MV2 MV2
m2run = 641π2fπ22 fπV2 lnMVA2 . (14) +α8dπ2M2f−π2M2 (cid:18)MA2 lnMλ22 − MV2 lnMλ22 (cid:19)(cid:21).
R A V A V
The picture laid down in [8] for the spectral function
in walking theories is the following: For scales below ForgivenvaluesofMV andMA,theS [8]andX param-
the continuum threshold which for s is O[4Σ(0)2], where etersarereducedinwalkingtheorieswithrespecttorun-
Σ(p) stands for the dynamical mass of the fermions, the ning theories (with α = 0). As long as MV and MA are
contributions are coming from distinct resonances like in smallerthaneλ2,whichduetoM2 <4Σ(0)2 <λ2(see
A/V
Π (s). FromthisscaleuptoΛ2,wherethesystemdrifts the more detailed explanation above) can be assumed,
run
awayfromconformalityagain,acontinuumofquarksand the pion mass is enhanced for walking theories.
3
y mˆ2 mˆ2 mˆ2
ΠUU ΠDD ΠUD
Walking technicolour. Let us study what this means 0 1 1 0.53
for TC theories. For these fπ2 = 2Λ2ew/Nfg where Λew = +1 2.87 1 1.47
246GeV. Ng stands for the number of techniflavours −1 1 2.87 1.47
f
gauged under the electroweak interactions. This number
need not be equal to the number of techniflavours. In a
situation where more than two techniquarks are needed TABLE I: Eigenvalues of mˆ2ab for the uneaten technipions in
MWT for various values of the parameter y.
to achieve walking dynamics it is advantageous to gauge
only two of them under the electroweak gauge group in
order to alleviate the pressure from the constraints on
masses are positive for all values of y. In Table I we
the oblique S parameter. This setup is known as par-
presentthenumericalvaluesforthemostnaturalchoices
tially gauged TC [3].
of y. g and g are weakly dependent on the mixing of
Apart from imposing the first Weinberg sum rule we 1 2
the electroweak gauge bosons with the composite vector
introduce a coupling g in the spirit of the coupling con-
statesinMWT.Weusetheirstandardmodel(tree-level)
nected to a local symmetry such that M4 = 2g2F2. It
V V values which is sufficient for our purposes.
is expected to be of order unity. For positive S there is
Thesquaresofthephysicalmassesas functionsof M
an upper limit for g above which F2 turns negative. In A
A and g are obtained by multiplying the values in Fig. 1
theplots,wedisplaym2 whichisamodelindependent
walk by the factors of Table I. Pion masses for the light mass
factor. The detailed mass structure mˆ2 differs between
models. Thenormalisationofm2 hasabbeenchosensuch window [6] of MWT, where MA < MV <∼ 1 TeV, are
walk from 50 to 300 GeV. For the heavy mass window, where
that the eigenvalues of mˆ2 are in general of order unity.
Hence, one can directly gaabin an initial impression on the MA > MV >∼ 2 TeV, all pion masses are at least 150
GeV. Interestingly, the pion masses can be large enough
magnitude of the pion masses from the numbers in the
toexceedpresentexperimentalboundsevenwithoutany
plots in Fig. 1: For small S parameters and large values
additionalextendedtechnicolour(ETC)interactions,ex-
of λ, M and g up to 400GeV can be reached, which
A pect in the region of small M and g.
corresponds to five times the mass of the W boson. A
Thelow-energylimitofthecontinuumisgivenbytwice
For a concrete model, MWT features two techni-
the dynamical mass Σ(0), i.e., the threshold of the loop.
quarks transforming under the adjoint representation of
It’s estimated to be 2Σ(0) ≈ 4πf /d . For MWT this
anSU(2)gaugegroup. Theadjointrepresentationisreal. π R
evaluates to circa 1 TeV. Picking resonances consider-
The unbroken chiral symmetry is therefore enhanced to
ably above this value entails a slight modification of the
SU(4) which breaks to SO(4) yielding nine Goldstone
picture, if one does not want to put them inside the con-
modes, three of which become the longitudinal degrees
tinuum: Thecontinuumthresholdhastobetakenhigher.
offreedomoftheweakgaugebosons. Thisleavesbehind
The only definite constraint for our investigation is that
six extra technipions which here carry nonzero baryon
the scale λ must lie inside the continuum. As we have
number. In this setup, the techniquarks alone would
chosen reference values of λ = 6 TeV and λ = 20 TeV,
lead to a theory with a global Witten anomaly, as an
we extend our plots to close to 3 TeV resonance masses.
odd number of fermions [The adjoint of SU(2) is three-
The model known under the name Next-to-Minimal
dimensional.] would transform under the SU(2) of the
L Walking Technicolour (NMWT) possesses two techni-
Standard Model. The anomaly is cancelled by including
quarks which transform under the two-index symmetric
anadditionalpairofleptons. Duetothepresenceofthese
representation of SU(3) which is not (pseudo)real, but
leptonstherequirementoffreedomfromgaugeanomalies
even-dimensional. Consequently, the flavour symmetry
does not fix the hypercharge assignment completely, but
breaking pattern is SU(2) ×SU(2) → SU(2) lead-
leaves one free continuous parameter. L R V
ing to three Goldstone modes which are all eaten by the
By using the generators of MWT from [3, 6] in (9)
weak gauge bosons. Hence, there are no degrees of free-
we find for the electroweak contributions to the masses
domleftwhichwouldreceivecontributionstotheirmass
of the technipions of MWT which are not eaten by the
in the framework of the present analysis.
electroweak gauge fields,
Other viable candidates for dynamical electroweak
(cid:0)g2+g2(cid:1)mˆ2 = g2(1+2y)2+g2 symmetrybreakingbywalkingTCtheoriesarediscussed
1 2 ΠUU 1 2 in [4] and can be analysed using the above formulae.
(cid:0)g2+g2(cid:1)mˆ2 = g2(1−2y)2+g2
1 2 ΠDD 1 2
(cid:0)g2+g2(cid:1)mˆ2 = g2(cid:0)4y2−1(cid:1)+g2. (20)
1 2 ΠUD 1 2 Summary. We studied the modifications of the Wein-
Here y is a parameter that controls the hypercharge as- berg sum rules in quasiconformal gauge field theories.
signmentofthetechniquarksandthesubscriptsU andD We showed that while the X parameter is not directly
denote the flavours of the techniquarks which constitute altered by nearly conformal dynamics, the electroweak
the corresponding technipion. The masses of the charge corrections to pion masses can be enhanced considerably
conjugate pions are the same. Since g >g the squared wrt. running theories. As an explicit example we dis-
2 1
4
√
FIG.1: Contourplotsofm2 asfunctionofM andg=M2/( 2F )forS =0(S =SMWT ≈0.16)andλ=6TeV(λ=20
walk A V V pert
TeV). Contours are for m2 = 0.0252 TeV2, 0.052 TeV2, 0.0752 TeV2, ...with zero at the origin. For S = SMWT, the plot
walk pert
range in g is constrained by the requirement F2 >0.
A
cussed MWT where the typical scale of these corrections TC directly the squared masses of the uneaten pions are
was seen to be from 100 to 200 GeV. Interestingly, the computed, the square root of which returns the linear
technipion masses can be large enough to exceed present masses. Therefore,itisunderstandablethatthethusob-
experimental bounds even without any additional ETC tained pion masses may (and do) exceed the values ob-
interactions, expect in the region of small M and g. tained by scaling up the QCD masses. Quasiconformal
A
The naive expectation that the masses (not the dynamics leads to an additional enhancement.
squared masses) at least in the running case can be sim-
ply scaled up from their value in quantum chromody-
namics to their value in techincolour by the ration say
of the respective pion decay constants is not confirmed. Acknowledgments. The authors acknowledge useful
For the QCD pions the above formulation returns the discussionswithRoshanFoadi,MadsT.Frandsen,Chris
difference if the squared masses of the charged and the Kouvaris, and Francesco Sannino. The work of DDD
neutral pions. In order to obtain the linear mass dif- was supported by the Danish Natural Science Research
ference, the result has to be divided by the sum of the Council. The work of MJ was supported by the Marie
masses, which is dominated by explicit symmetry break- CurieExcellenceGrantundercontractMEXT-CT-2004-
ing, i.e., the bare quark masses. In the application to 013510.
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