Table Of ContentWW Scattering Parameters via Pseudoscalar Phase Shifts
T. Appelquist,1 R. Babich,2 R. C. Brower,2 M. I. Buchoff,3 M. Cheng,3 M. A. Clark,4
S. D. Cohen,5 G. T. Fleming,1 J. Kiskis,6 M. F. Lin,1 E. T. Neil,7 J. C. Osborn,8
C. Rebbi,2 D. Schaich,2,9 S. Syritsyn,10 G. Voronov,1 P. Vranas,3 and J. Wasem3
(Lattice Strong Dynamics (LSD) Collaboration)
1DepartmentofPhysics, SloaneLaboratory, YaleUniversity, NewHaven, Connecticut06520, USA
2Department of Physics, Boston University, Boston, Massachusetts 02215, USA
3PhysicalSciencesDirectorate,LawrenceLivermoreNationalLaboratory,Livermore,California94550,USA
4Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
5Department of Physics, University of Washington, Box 351560, Seattle, WA 98195, USA
6Department of Physics, University of California, Davis, California 95616, USA
7TheoreticalPhysicsDepartment, FermiNationalAcceleratorLaboratory, Batavia, IL60510, USA
8Argonne Leadership Computing Facility, Argonne, Illinois 60439, USA
2 9Department of Physics, University of Colorado, Boulder, CO 80309, USA
1 10Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
0
Usingdomain-walllatticesimulations,westudypseudoscalar-pseudoscalarscatteringinthemaximal
2
isospinchannelforanSU(3)gaugetheorywithtwoandsixfermionflavorsinthefundamentalrepresen-
n tation. ThiscalculationoftheS-wavescatteringlengthisrelatedtothenext-to-leadingordercorrections
a
toWWscatteringthroughthelow-energycoefficientsofthechiralLagrangian. Whiletwoandsixflavor
J
scattering lengths are similar for a fixed ratio of the pseudoscalar mass to its decay constant, six-flavor
9
scatteringshowsasomewhatlessrepulsivenext-to-leadingorderinteractionthanitstwo-flavorcounter-
1
part.EstimatesaremadefortheWWscatteringparametersandtheplausibilityofdetectionisdiscussed.
]
t PACSnumbers:11.10.Hi,11.15.Ha,11.25.Hf,12.60.Nz
a
l
-
p INTRODUCTION electroweak symmetry breaking, WW scattering could be
e
animportantchanneltoinvestigateattheLHC.
h
[ Technicolor theories based on scaled-up QCD may be
disfavoredduetolargecontributionstotheelectroweakS-
1 Ourapproachistocomputetheparametersofeffective,
v parameterandachiralcondensatetoosmalltoaccountfor low-energy chiral Lagrangians. We focus on two coeffi-
7 flavor effects [1]. However, walking [2–4] could account
cientsoftheelectroweakchiralLagrangian[12,13]which
7 forthesedifferences. Foragaugetheorywithafixednum-
encode the dominant deviations of the longitudinal WW
9 ber of colors, walking is believed to set in at a number of
3 scattering amplitude from that of the standard model [14–
flavors N just below the minimum number Nc required
. f f 16]. Thesecoefficients,inturn,arerelatedtocertaincoeffi-
1 to generate an infrared fixed point. Above Nc, the theory
0 isconformalintheinfrared,andthisbehaviorfpersistsuntil cientsoftheNf-flavorhadronicchiralLagrangian[17,18].
2 ForQCD(N = 2),therearetwosuchparameters(l and
asymptoticfreedomislost. f 1
1 l ). Lattice methods have so far constrained only linear
v: Sincewalkingandtheonsetofinfraredconformalityin- c2ombinations of these terms via pion form factors [19] or
volvestrongdynamics,numericallatticefieldtheoryiside-
i theextractionofeffectiverangeparametersinI=2ππscat-
X ally suited to study this behavior. To this end, lattice cal-
tering[20].
r culations [5–9] have been performed with different num-
a
bers of flavors, colors, and in different representations,
leading to evidence for both confining and conformal be- Herewecalculatetheleadingterm(scatteringlength)in
havior, with much debate as to the boundaries of these theeffective-rangeexpansionforS-wave,maximal-isospin
phases. The present work builds on two earlier papers of pseudoscalar scattering for the N = 2 and N = 6 the-
f f
the LSD collaboration [10, 11], where evidence for both ories. Forthelinearcombinationofchiral-Lagrangianpa-
chiral condensate enhancement and dynamic reduction of rametersenteringthescatteringlength,thisprovidesafirst
theS-parameterwereobservedforanSU(3)gaugetheory glimpseofhowthereducedrunningofthecouplingassoci-
whenN isincreasedfrom2to6,stillwellbelowNc. atedwiththeincreaseofN affectsthedynamics. Insub-
f f f
In this work, using domain-wall fermions, we make the sequentwork,wewillcomputeadditionalchiral-parameter
first lattice connection to WW scattering for these two combinationsthroughbothD-waveprojectionsandfurther
theories. This process, whose longitudinal modes can be effective range parameters. Together, these will lead to a
related to scattering of pseudo-Nambu-Goldstone bosons prediction for the two parameters of the electroweak chi-
(PNGBs) via the equivalence theorem, is receiving re- ral Lagrangian describing strongly coupled corrections to
newed attention. As a direct probe of the physics behind WWscattering.
2
ELECTROWEAKCHIRALLAGRANGIANANDWW
SCATTERING Hadronic EW
EFT EFT
The electroweak chiral Lagrangian (EWχL) allows for
asystematicdescriptionofelectroweakscaleeffectsresult-
ing from TeV scale physics [12, 13]. This effective chiral md →0 g,g →0
Lmaegtrrya.ngHiaonwmevuesrt,rtehsepecchtirthaleLSaUgr(a2n)gLi×anUa(p1p)rYoxgimauagteelsyymre-- p2 Md2s,Ms2s p2 Md2s,Ms2s
spectsalarger“custodial”symmetrySU(2)L×SU(2)C, f22tr(∂µU†∂µU)+α5 tr(∂µU†∂µU) 2+α4 tr(∂µU†∂νU) 2
which spontaneously breaks to the diagonal subgroup.
From the EWχL[12, 13], the dominant contributions to
FIG.1:
WWscatteringcomefromtheterms
ComparisonbetweenlimitsofthehadronicchiralLagrangian
W4 = g2tr[Wµ,Wν]2 withNf flavors[17,18]andtheelectroweakchiralLagrangian
L − [12,13].Symbolicquantitiesaredefinedinthetext.
+2ig tr (∂ W ∂ W )[W ,W ]
µ ν ν µ µ ν
−
+α [tr((cid:16)V V )]2+α [tr(V Vµ)]2, (cid:17) (1)
4 µ ν 5 µ To exhibit the flavor-dependent dynamics giving rise to
α and α , an extension to the multi-flavor hadronic chi-
where Vµ = (DµU)U†, U(x) is the unitary matrix field 4 5
ralLagrangian,withnon-zerofermionmassesisappropri-
that transforms under SU(2) SU(2) (akin to the
L C
× ate. TheorieswithmultipleflavorsleadtoadditionalGold-
hadronic matrix of Goldstone fields), and the covariant
stonedegreesoffreedom,uneatenbytheW andZ bosons,
derivativeofU(x)isgivenby
which,withasmallmass,becomePNGBs. Theircontribu-
(cid:126)τ τ tiontophysicalphenomenaareparametrizedbythemany
D U = ∂ U +ig W(cid:126) U ig U 3B . (2)
µ µ 2 · µ − (cid:48) 2 µ low-energyconstants(LECs)ofthemulti-flavorchiralLa-
grangian. The presence of a finite fermion mass is also
Theα andα termsdescribe (g4)correctionstoWW
4 5 O essentialforthelatticesimulationsemployedhere.
scattering.
For theories with N 4 massive fermions, there are
In Ref. [21], Eboli et al found that with an integrated f ≥
9 LECs in the NLO hadronic chiral Lagrangian, denoted
luminosity of 100 fb 1, and by considering both W
−
byL [18]. TheLECsL multiplytermsproportional
and Z scattering, the LHC could place 99% CL bounds 0 8 4 8
to the−fermion mass, while−L multiply terms that are
7.7 10 3 < α < 15 10 3 and 12 10 3 < 0 3
− × − 4 × − − × − independentofthefermionmas−s. Onewaytorelatethese
α < 10 10 3. BecausethefittopotentialLHCdatawas
5 × − LECs to the EWχLis by assigning electroweak quantum
made using only the terms of Eq. (1) at tree level, the α-
numbers to one fermion doublet among the N fermions,
parametersare,ineffect,definedatalowscale,incorporat- f
leavingtheothersneutral.
ingallradiativecorrectionsincludingbothstandard-model
Although our lattice simulations are carried out with a
corrections and new physics. These parameters were also
common mass m for all the fermions, it is helpful for our
constrained by unitarity considerations in Refs. [15, 16].
discussiontotemporarilyassignamassm tothefermion
Other custodial-symmetry respecting coefficients in the d
doublet with electroweak quantum numbers and a mass
EWχLhave been constrained experimentally and do not
m m totheremainingelectroweaksinglets. Thefor-
lead to appreciable corrections to WW scattering [15]. s d
≥
mer must be taken to zero, while the latter may or may
MorerecentassessmentsofLHCconstraintsonvectorbo-
not be taken to zero. The N2 1 PNGBs are then sep-
sonscatteringwereperformedinRef.[22]andRef.[23]. f −
arated into the 3 (with mass denoted by M ) which be-
Inthelimitg,g(cid:48) 0,theEWχLreducestothemassless dd
→ comemasslesswhenm 0andareeatenbythegauge
two-flavorhadronicchiralLagrangian[17],asillustratedin d
→
bosons, the 4(N 2) composed of one electroweak-
Fig.1,where f
−
doubletfermionandoneelectroweak-singletfermion(with
α5 = (cid:96)41 +O(g) monalsysedleecntortoewdebayk-Msindgsl)e,tafnedrmthioen(sN(wfi−th2m)a2scsodmenpootseeddboyf
(cid:96) M ). All except the 3 eaten modes can also get masses
α = 2 + (g), (3) ss
4 4 O from standard-model and beyond-standard-model interac-
tionsnotincludedhere.
where (cid:96) and (cid:96) are the next-to-leading-order (NLO)
1 2 The relation of the 9 NLO LECs of the general-flavor
Gasser-Leutwyler coefficients of the hadronic chiral La-
hadronic Lagrangian to α and α in the electroweak La-
grangianwithonlyderivativecouplings, 4 5
grangian is mapped out in Fig. 1. “High-energy physics”
(cid:96) (cid:96) including the PNGB particles composed of one or two of
L(cid:96)N1L(cid:96)2O = 41[tr(∂µU†∂µU)]2+ 42[tr(∂µU†∂νU)]2. (4) theelectroweak-singletfermionsofmassmsareintegrated
3
out to describe observed low-energy physics on the order the second ( 1/192π2)[log(M2 /M2) + O(1)], where
− dd H
oftheW andZ masses. Specifically,itisconventionalto the O(1) constants are determined by the aforementioned
start with a set of 9 LECs Lr(µ) defined at some scale µ one-loopcalculation. Theremainingquantitiesarethen
i
small compared to the breakdown scale of the chiral La- lr(µ,M ) log MH2 +O(1)
grangian. They are independent of PNGB masses, being α (M ,M ) = 1 ds µ2
definedinthezero-masstheory,andinfraredfinitebecause 5 H ds 4 − (cid:2) 384π2 (cid:3)
ofthepresenceofthescaleµ. lr(µ,M ) log MH2 +O(1)
Now suppose that the two-flavor Lagrangian of Eq. (4) α(cid:101) (M ,M ) = 2 ds µ2 .
4 H ds 4 − 192π2
refers to the two electroweak-doublet fermions. The two (cid:2) (cid:3)
(7)
coefficients, lr and lr, can also be defined at some scale
1 2 (cid:101)ThesubtractedLECsα (M ,M )andα (M ,M )
µ, and in the limit M = 0. Since the PNGBs of mass 4 H ds 5 H ds
dd aredirectlyanalogoustotheconventionallydefinedS pa-
M andM havebeen“integratedout”(integratedintolr
ds ss 1 rameter,asemployedinRef.[11]. Theycanbedetermined
and lr) these two coefficients do depend on these masses. (cid:101) (cid:101)
2 experimentallyandcomparedtolatticecomputationsifthe
At NLO and with the PNGB masses small compared to
additional,standard-modelcorrections,withthesameM ,
the breakdown scale, they are related to the Lr(µ) by the H
i arealsotakenintoaccount. ForN > 2,theyremainsen-
additionofchirallogarithmsarisingfromaloopofPNGBs f
sitive to the PNGB masses M , and therefore the limit
composedofoneEW-doubletfermionandoneEW-singlet ds
m 0canbetakenonlyifadditionalphysicsprovidesa
fermion[18,24]: s
→
massforthesemodes.
(cid:96)r(µ,M ) = 2Lr(µ)+4Lr(µ)+2Lr(µ)
1 ds − 0 1 3
2 N M2
+ − f log ds LATTICEFRAMEWORK
24(32π2) µ2
(cid:96)r2(µ,Mds) = 4Lr0(µ)+4Lr2(µ) For the lattice calculation being performed here, all Nf
2 N M2 fermionsareassignedacommonmassm,and,asaresult,
+ − f log ds. (5)
haveacommonPNGBmassM . Thegoalistocompute
12(32π2) µ2 P
the resultant Lr(µ) through pseudoscalar scattering, and
i
The coefficients of the the chiral logarithms are propor- then make use of Eqs. 5-7 with M = M = M , to
P ds dd
tionaltotheexpectedcountingfactorof4(Nf −2). determinethequantitiesα4(MH,MP)andα5(MH,MP)
These quantities are related to the corresponding asafunctionofM . ForN = 2,thelimitM 0isfi-
P f P
electroweak LECs α5r(µ,Mds) and α4r(µ,Mds) by nite. ForNf > 2,theres(cid:101)iduallogMP depe(cid:101)nden→cemeans
Eq. (3). Then, a set of µ-independent electroweak thatthelimitm 0canbetakenonlyifinteractionsnot
LECs α5(Mdd,Mds) and α4(Mdd,Mds) can be defined includedheregen→erateamassforthesemodes.
by adding chiral logarithms arising at NLO from a loop
of PNGBs composed of electroweak-doublet fermions of
(cid:101) (cid:101)
massm : MAXIMAL-ISOSPINPSEUDOSCALARSCATTERING
d
lr(µ,M ) 1 M2
α (M ,M ) = 1 ds log dd The study of pion scattering is well established in ef-
5 dd ds 4 − 384π2 µ2
fective field theory, where the leading order (LO) result
α(cid:101) (M ,M ) = l2r(µ,Mds) 1 log Md2d,(6) wasfirstcalculatedbyWeinberg[25]andtheNLOchiral-
4 dd ds 4 − 192π2 µ2 perturbation-theory result was calculated by Gasser and
Leutwyler[18]. Ourfocushereisonthescatteringoftwo
whe(cid:101)re we have dropped (g) corrections. These elec- identicalpseudoscalarswithinanSU(2) SU(2) sub-
O L R
troweak LECs have incorporated into them all strong- ×
group of the flavor symmetry, taken here to refer to the
interaction effects including the PNGBs made purely of
fermion doublet assigned electroweak charges. This pro-
EW-doubletfermions,givenatNLObytheexplicitchiral-
cess is often referred to as “I=2” scattering in the case of
logarithmterms.
twoflavors. Here,werefertoitas“maximal-isospin”scat-
Comparisonwiththeα andα employedbyEbolietal
4 5 teringforN fermionflavorswithdegeneratemasses.
f
[21] requires the inclusion of standard-model corrections
Low-energy pseudoscalar scattering can be parameter-
notconsideredhere. Thentheabovelog(M2 )dependence
dd ized in terms of the phase shift, δ. On the lattice, the ac-
can be separated and incorporated into these corrections
cessible quantity is (cid:126)k cotδ, where (cid:126)k is the magnitude
(allowing the limit Mdd → 0 to be taken there) since the ofthepseudoscalar3|-m|omenta. TheS|-|waveprojectionof
resultant NGBs are eaten by the W and Z. We therefore
this combination has a convenient effective-range expan-
subtract from Eq. (6) the one-loop contributions arising
sionforsmallmomentagivenby
from PNGBs of mass M together with a Higgs boson
with reference mass MH.ddFrom the first of Eq. (6), we (cid:126)k cotδ = 1 + MP2rPP |(cid:126)k|2 + |(cid:126)k|4 , (8)
subtract ( 1/384π2)[log(M2 /M2) + O(1)], and from | | a 2 M2 O M4
− dd H PP (cid:18) P(cid:19) (cid:18) P(cid:19)
4
where a is the scattering length (in the particle physics (5) vanish. From Eq. (7), we then have for the N = 2
PP f
convention), r is the effective range, and M is the case
PP P
pseudoscalarmass. Theexpansionisvalidfor (cid:126)k MP. α (M )+α (M ) = 1 br (µ)
The NLO chiral expansion for the scatterin|g|le(cid:28)ngth in 4 H 5 H −512π2 PP
theMIchanneltakestheform[26,27] log MH2 +O(1)
(cid:101) (cid:101) µ2 . (14)
M2 M2 − (cid:2) 128π2 (cid:3)
MPaPP = −16πF2(cid:40)1+ 16π2F2(cid:34)brPP(µ) Thus, for Nf = 2, a lattice calculation of MPaPP as a
functionofM,withafitusingEq. (9),candirectlydeter-
− 2(NNff−2 1) +A(Nf)log(cid:16)Mµ22(cid:17)(cid:35)(cid:41), (9) mtakiFneoenrα(cid:101)tNo4(fNM=NHL2)O,+iotriα(cid:101)ds5ea(rlMsionHwM)o.r2t/hFn2o,tiαng4 athnadtαif5thceananbaelydseitseirs-
minedseparately,employingonlythequantitiesa ,M
where M2 = 2m ψψ /F2, with F and ψψ the pseu- PP P
(cid:104) (cid:105) (cid:104) (cid:105) andF . TheNLOLECsintroducedabovemultiplysingle
doscalar decay constant and chiral condensate at m = 0, P (cid:101) (cid:101)
chiral-logarithm terms at this order. While promising, the
andwhere
primary difficulty is having enough data within the chiral
(2 N +2N2+N3)
A(N ) = − f f f , regimetoconstrainthesetermsaccurately.
f N2
f
br (µ) = 256π2 (N 2)(Lr(µ) Lr(µ))
PP − f − 4 − 6 LATTICEDETAILS
+Lr(µ)+2Lr(µ)+2Lr(µ)+Lr(µ) .
0 (cid:2) 1 2 3
(10) Lattice calculations are performed using 323 64
(cid:3) ×
domain-wall lattices with the Iwasaki improved gauge ac-
EachLr(µ)isdefinedatascaleµintheMS scheme. The
i tion with a fifth dimensional length of Ls = 16 and a
quantitybr (µ)+A(N )log(M2/µ2)isµ-independent.
PP f domain-wall height of m0 = 1.8 [10, 11]. By using
Similarchiralexpansionsofthepionmassandpionde-
domain-wall fermions, the calculation preserves exact fla-
cayconstantaregivenby[27]
vor symmetry and chiral-breaking lattice spacing artifacts
are L suppressed (m m). This is performed for
M2 1 M2 s res (cid:28)
MP2 = M2(cid:40)1+ 16π2F2(cid:34)brM(µ)+ Nf log µ2 (cid:35)(cid:41) Nbefta=va2luaetsβwe=re2c.h7o0seanndtoNmfat=ch6thaetIβR=sca2l.e1s0o,fwbhoethreththee-
ories and correspond to an inverse lattice spacing roughly
M2 N M2
FP = F(cid:40)1+ 16π2F2(cid:34)brF(µ)− 2f log µ2 (cid:35)(cid:41), Nfive=tim6e,sfithvee vmeacstsorpmoienstsonarmeaasnsa.lyFzoerdbwoitthhNmf ==m2 an+d
f f
m ,wherem = 0.010,0.015,0.020,0.025,0.030.
(11) res f
where
FINITEVOLUMEMETHOD
br (µ) = 128π2[N (2Lr(µ) Lr(µ))
M f 6 − 4
+2Lr(µ) Lr(µ)]
8 − 5 In Euclidean space, scattering phase shifts can be ex-
br(µ) = 64π2[N Lr(µ)+Lr(µ)]. (12) tracted on the lattice only by calculating the total energy
F f 4 5
of two hadrons in a finite volume [28]. In practice, this
TheexpressionsforM andF arealsoµ-independent.
P P is accomplished by calculating the four-point correlation
Only the LECs L -L contribute to WW scattering,
0 3 functionandexploringthelongtimebehavior
(Eq. (5)), since only they multiply operators that survive
asm 0. Butin generalthe scatteringlength computed tr(π(t)2) tr(π(t))2 e−EPPt, (15)
→ − ∼
herewillgiveusthecombinationofparametersappearing where the zero-momentum projected operator π(t) is re-
in br (Eq. (10)), which include L - L . More data is lated to the pseudoscalar two-point function by C (t) =
PP 4 8 P
requiredtoseparatethesesetsforgeneralNf. tr(π(t)), and E2 = 4((cid:126)k 2 + M2), with (cid:126)k being the
ForNf = 2,however, center-of-mass sPcPattering|m|omentuPm. In th|is|work, we
restrict ourselves to S-wave scattering by projecting each
br (µ) = 256π2 Lr(µ)+2Lr(µ)
PP − 0 1 pseudoscalar correlator onto zero momentum. For a finite
+(cid:2)2Lr2(µ)+Lr3(µ) volume, only discretized values of the scattering interac-
= 128π2 lr(µ)+lr(µ) . (13) tion are allowed. This scattering momentum is related to
− 1 2 (cid:3) thephaseshiftby[28]
Both lr(µ) and lr(µ) are n(cid:2)ow M -indep(cid:3)endent since
1 2 P 1 (cid:126)k 2L2
the only PNGBs are the three composed of EW-doublet (cid:126)k cotδ = S | | , (16)
fermions, and therefore the chiral-logarithm terms in Eq. | | πL (cid:32) 4π2 (cid:33)
5
where the function S(η) is given by the regularized zeta 2
function[29]
1.5
Λ 1
S(η) = 4πΛ. (17)
j 2 η − m
(cid:88)j(cid:54)=0 | | − / 2 1
2P
M
N=2 LO+NLO
f
CALCULATIONANDFITTING 0.5 N=2
f
N=6
f
The calculation yields two quantities: the four-point
pseudoscalar correlation function, C (t), and the two- 0
PP 0 0.01 0.02 0.03
point correlation function C (t). The F values used in m
P P 0.05
thisworkwerepreviouslycalculatedinRef.[10]. Thelong
timebehaviorofthetwocorrelatorsdifferduetotheextra
backward propagating pseudoscalar that exists in C (t) 0.04
PP
and as a result, the long time behavior of the midpoint-
symmetriccorrelatorsisgivenby 0.03
P
F
C (t) A+Bcosh(E t)
PP PP 0.02
→
C (t) Ccosh(M t). (18) N=2 LO+NLO
P P f
→ N=2
0.01 f
N=6
Forthepseudoscalarmass,ahyperboliccosinefitwasper- f
formed,where
0
0 0.01 0.02 0.03
m
C (t+1)+C (t 1)
cosh(M ) = P P − . (19)
P 2CP(t) 0
AdifferentfittingformwasusedforEPP,namely ot δ
c -100
C (t+2) C (t 2) |→k
cosh(EPP) = 2(CPPPP(t+1)− CPPPP(t− 1)). (20) /| m
− −
MP-200 N=2 LO+NLO
f
Constantfitswereperformedtotheinversehyperbolicco- N=2
f
sine of Eq. (19) and Eq. (20) over a window of roughly N=6
f
40timeslicesintheplateauregion. Inadditiontostatisti-
-300
calerrors,systematiceffectsduetoplacementofthefitting 0 0.005 0.01 0.015 0.02 0.025 0.03
m
windowwerealsoexaminedbyvaryingthewindowby 2
±
timeslicesoneachside.
FIG. 2: Plots of M2/2m, F and M /m|(cid:126)k|cotδ (cid:39)
P P P
M a /m versus m (in lattice units). The error bars are sta-
P PP
tistical plus systematic. The red circles represent the N = 2
f
dataandthebluesquaresrepresenttheN =6data. Thefitsfor
SCATTERINGRESULTS f
N =2aremadeusingonlythesolidredpoints.
f
In our lattice calculations, (cid:126)k cotδ is extracted from
| |
Eq. (16). For Eq. (9) to be applicable, the effective range
expansion, Eq. (8), is needed and (cid:126)k 2/M2 must be suf- Nf =2
| | P
ficiently small. The data suggest that the approximation
M /(cid:126)k cotδ M a is valid within 9%, and thus Todeterminebr (µ)forN = 2, wecarryoutacom-
P | | ≈ P PP PP f
Eq. (9) should be applicable to the lattice results. Our bined fit using NLO chiral perturbation to M a (Eq.
P PP
results for pseudoscalar scattering in the MI channel are (9)), M2 (Eq. (11)), F (Eq. (11)), and the chiral con-
P P
given as a function of m in the bottom graph of Fig. 2. densate. Thelatterthreequantitieswerecomputedandan-
We first analyze the N = 2 results and then discuss the alyzed in Ref. [10], where it was concluded that for the
f
differencebetweenN = 2andN = 6. range of m values employed there, NLO chiral perturba-
f f
6
tion theory provides an acceptable fit allowing a reliable setofO(g4)contributionswillfallcomfortablywithinthe
extrapolationtom = 0,determingtheextrapolateddecay Eboli et al bounds based on 100 fb 1 of LHC data. Fu-
−
constantF andchiralcondensate ψψ . turemeasurementswillhavetobemoreprecisetocompare
(cid:104) (cid:105)
meaningfullywithEq. (22).
TABLE I: Results (in lattice units) of a combined NLO fit for
N =2toM a ,M2,F ,andthechiralcondensate.
f P PP P P N =6
f
mf=0.01–0.015 mf=0.01–0.02 mf=0.015–0.02 Ref.[10]
brC(F) 80(10) 110.7(9.0) 198(60) 91(29) The Nf = 6 data for MP/m(cid:126)k cotδ MPaPP/m
brF(F) 5.20(28) 5.38(15) 5.09(58) 5.70(27) displayed in Fig. 2 lies statisticall|y|on top(cid:39)of the N = 2
br (F) -2.36(21) -1.74(24) 0.5(2.2) -2.22(62) f
M data. A NLO fit using Eq. (9) would nevertheless lead to
br (F) -17.32(88) -16.89(59) -13.3(3.8) —
PP a different value for br (µ) because the chiral-logarithm
F 0.0220(16) 0.0229(10) 0.0262(35) 0.0209(41) PP
terminthisexpressionhasamuchlargercoefficient,grow-
ψψ /F2 1.049(74) 0.885(39) 0.65(10) 0.99(17)
χ(cid:104)2/d(cid:105)of 16/2 83/6 13/2 26/4 inglinearlywithNf. However,thelargercoefficientindi-
cates that it is very unlikely that Eq. (9) can be employed
in the N = 6 case for the existing range of m values.
For the present fit, we choose µ = F 250 GeV (In f
Ref.[10], thescaleµ = 4πF wasused.) (cid:39)The(combined) This point was already made in Ref. [10] where FP, MP
and the chiral condensate were computed. The NLO ex-
fits for M2/2m, F and M /m(cid:126)k cotδ M a /m
P P P | | (cid:39) P PP pressionforFP (Eq. (11))alsohasachiral-logarithmterm
are shown in Fig. 2. Only the points m = 0.01 0.02
f withacoefficientgrowinglinearlywithN .
− f
(shown in solid red) are used in the fit. For all quantities,
Nevertheless, our results for N = 6 do indicate an in-
f
NLO chiral perturbation theory again provides an accept-
teresting trend, which can be seen more easily by plotting
ablefitasinRef.[10].
thedatainadifferentway. IftheNLOchiralexpansionEq.
The fit parameters are shown in bold face in the central
(9) is reorganized in terms of the physical values M and
P
columnofTableI.Inaddition,fitparametersfortwoother
F ,ittakestheform
P
m ranges are shown, and used to estimate systematic er-
f
rors. Fromthetable,thequantityofinteresthereis M2 M2
M a = P 1+ P br (µ)
brPP(µ = F) = −128π2 l1r(µ = F)+l2r(µ = F) P PP − 16πFP2(cid:40) 16π2FP2(cid:34) P(cid:48)P
(cid:2) (cid:3) 2(N 1) M2
= −16.89±0.59+−30..5493 dχo2f = 863, − Nff−2 +A(cid:48)(Nf)log(cid:16) µ2P(cid:17)(cid:35)(cid:41),
(21) (23)
where we have used the central value from the fit range where
m = 0.01 0.02. Theerrorsarestatisticalplussystem-
f 2(1 N +N2)
atic. This re−sultis consistent withprevious lattice simula- A(N ) = − f f
(cid:48) f N2
tionsofI=2pion-pionscattering[20,30–35],other2+1lat- f
ticeQCDdeterminationsoftheseLECs[19,20],andQCD br (µ) = 256π2 Lr(µ)+2Lr(µ)+2Lr(µ)
phenomenology. Lighter mass ensembles will be required (cid:48)PP − 0 1 2
+Lr(µ) 2Lr(µ) Lr(µ)+2Lr(µ)
forfuturecalculationstoachievehigherprecision. 3 (cid:2)− 4 − 5 6
FromEq. (14),wethenhave,forN = 2, +Lr(µ) . (24)
f 8
α4(MH)+α5(MH) = (3.34±0.17+−00..0781)×10−3 WlogitahriMthmPiacPtPeremxprreemssaeidn(cid:3)sinfitnhiitsewasayN, theinccoreefafisceise,natnodftthhee
(cid:101) (cid:101) log MFH22 +O(1) , (22) LEC combination b(cid:48)PrP(µ) contains noFexplicit Nf depen-
− 128π2 dence.
(cid:2) (cid:3)
where the errors are again statistical plus systematic. Re- Theaboveexpansion,whilenotapriorimorereliablefor
callthattheexpressioninthesecondlinearisesfromhav- Nf = 6 than Eq. (9), suggests that a plot of MPaPP
(cid:39)
ing removed the one-loop contribution arising from the M /(cid:126)k cotδ versus the physical quantity M2/F2 could
P | | P P
eatenGoldstonebosonsandaHiggsbosonwithreference be revealing. This is done in Fig. 3, where the solid-color
mass M . For a wide range of M , it is O(10 3) or pointscorrespondtotherangem = 0.01 0.02. Asmall
H H − f
−
smaller. upward shift of the N = 6 points relative to N = 2 is
f f
Once the additional one-loop standard-model correc- indicated. Since a negative value for a corresponds to
PP
tionswiththesameM areincluded,theaboveresultcan a repulsive interaction, the data indicate that the N = 6
H f
becompareddirectlywithLHCdata. Ifthestandard-model theoryissomewhatlessrepulsivethantheN = 2theory
f
correctionsarealsooforder10 3 orsmaller, thenthefull forpseudoscalarscatteringinthemaximal-isospinchannel.
−
7
0 0
N=2
f
-0.2 N=6
f
-2
δ -0.4
→||k cot -0.6 -10229 a) -4
/ 0.
MP-0.8 µ = -6
-1 LNO=2 r′b(PP
f
N=6
f -8
-1.2
0 10 20 30 40
2
( M / F )
P P -100 0.005 0.01 0.015 0.02 0.025 0.03
m
FIG.3: PlotofM /|(cid:126)k|cotδ (cid:39) M a vs. (M /F )2. The
P P PP P P
errorbarsarestatisticalplussystematic.Theredcirclesrepresent FIG.4: Chiralparameterb(cid:48)PrP versusfermionmassmforNf =
thetwo-flavordataandthebluesquaresrepresentthesix-flavor 2andNf =6.
data. ThedashedlineistheLOχPTresult(zeroparameterfit).
Largernegativeresultscorrespondtomorerepulsivescattering.
It is not yet clear whether this fit can be trusted for
N = 6, but even if it can, the resultant value for
f
The dashed line, representing the LO expression br (µ = 0.023a 1 F) determines only the combi-
M2/16π2F2, is a reasonably good first approximation nPa(cid:48)tPion of LECs in−Eq.(cid:39)(24), which includes Lr(µ) values
− P P i
tothedataforbothN = 2andN = 6. ForN = 2,the not directly relevant to WW scattering. Further calcula-
f f f
data show that the effect of the NLO term is to make the tions will be necessary to isolate α (M ,M = M )
4 H P ds
interactionmorerepulsive. Thequantityinsquarebrackets and α (M ,M = M ) (Eq. (7)). This will then de-
5 H P ds
in Eq. (23) is positive and of order unity within the range scribe the effect of beyond-standard-model physics for a
(cid:101)
shown. A fit to just M a with µ = F leads to the rangeofPNGBmassesM .
P PP (cid:101) P
value br (µ = F) = 4.67 0.65+1.06. Clearly there
issome(cid:48)PcPancelationbetw−eenth±isterm−a0n.d05thechiralloga-
rithm. Nonetheless, this br value (when combined with SUMMARYANDDISCUSSION
(cid:48)PP
thebr andbr valuesinTableI)isconsistentwiththebr
M F PP
valueinEq.(21). Using lattice simulations, we have computed
For N = 6, the data is even closer to the LO dashed pseudoscalar-pseudoscalar scattering in the maximal
f
line, suggesting that NLO perturbation theory in the form isospin channel for an SU(3) gauge theory with two and
of Eq. 23 might again be reliable. If this expression is six fermion flavors in the fundamental representation.
used to fit the N = 6 data, then the quantity in square Our calculation of the S-wave scattering length was then
f
brackets is again positive and of order unity within the related to the next-to-leading order (NLO) corrections
rangeshown, butsomewhatsmallerinmagnitudethanfor to WW scattering through the low-energy coefficients
N = 2. Since we don’t yet know the precise value of F of the chiral Lagrangian. For N = 2, our result for
f f
inlatticeunitsforN = 6,wecarryouttheNLOfitusing the scattering length agreed with previous calculations,
f
thescaleµ = 0.023a−1 (F forNf = 2). Thefitleadsto showing an increase in repulsion due to the NLO correc-
br (µ = 0.023a 1 F) = 7.81 0.46+1.23, larger tions. For WW scattering, we obtained an estimate for
in(cid:48)PPmagnitudethan−for(cid:39)N = 2.−Therei±snowm−0o.5re6 cance- α (M )+α (M )(Eq. (22))describingdeviationsfrom
f 4 H 5 H
lation between this term and the chiral logarithm than for thestandardmodel.
N = 2. Six-flavor scattering showed a somewhat less repulsive
f (cid:101) (cid:101)
The above values of br emerge from a fit of Eq. (23) NLOinteractionthanitstwo-flavorcounterpartforafixed
(cid:48)PP
to each of the three lightest data points (corresponding to ratio of the pseudoscalar mass to its decay constant. The
m = 0.01 0.02),withafixedchoiceµ = 0.023a 1 rangeoffermionmassesemployedsofardoesnotallowa
f −
− (cid:39)
F. Aplotoftheresultantvalueofbr versusm(Fig. 4), clearlyreliableuseofchiralperturbationtheory. Also,the
(cid:48)PP
shows that br (µ = 0.023a 1 F) is relatively inde- appearanceofmoretermsinthehadronicchirallagrangian
(cid:48)PP − (cid:39)
pendent of m for both N = 2 and N = 6 as expected for six flavors does not allow the extraction of only the
f f
if NLO perturbation theory is reliable. The evident shift combination of parameters entering WW scattering. Fur-
going from N = 2 to N = 6 is interesting since this thersimulationsofadditionallow-energyscatteringparam-
f f
quantity is contains LEC’s that enter into WW scattering etersatlowerfermion-massvalueswillberequiredtocom-
throughEq. (24). pletethisstudy.
8
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