hadronic matrix elements for dark matter and other searches ...laurent lellouch cpt marseille cnrs...
TRANSCRIPT
Hadronic matrix elements for Dark Matterand
other searches
Laurent Lellouch
CPT MarseilleCNRS & Aix-Marseille U.
Budapest-Marseille-Wuppertal collaboration (BMWc)
(Phys.Rev.Lett. 116 (2016) 172001 and in preparation)
Laurent Lellouch KEK-PH, , 13-16 February 2017
Direct WIMP dark matter detection
↓χ χ
N N
H
Lqχ =∑
q
λΓq[qΓq][χΓχ] −→ LNχ = λΓ
N [NΓN][χΓχ]
Quark are confined within nucleons→ nonperturbative QCD tool
Laurent Lellouch KEK-PH, , 13-16 February 2017
WIMP-nucleus spin-independent cross section . . .In low-E limit
dσSIχ A
Z X
dq2 =1πv2 [Zfp + (A− Z )fn]2 |FX (q2)|2
w/ FX (~q = 0) = 1 nuclear FF and χN couplings (N = p, n)
fNMN
=∑
q=u,d,s
f Nqλq
mq+∑
Q=c,b,t
f NQλQ
mQ
such that (f = u, . . . , t and 〈N(~p′)|N(~p)〉 = (2π)3δ(3)(~p′ − ~p))
f Nud MN = σπN = mud〈N|uu + dd |N〉, f N
f MN = σfN = mf 〈N |f f |N〉
For heavy Q = c, b, t (Shifman et al ’78)mQQQ
−→ mQ QQ = −13αs
4πG2 + O
(α2sO6
4m2Q
)
Laurent Lellouch KEK-PH, , 13-16 February 2017
. . . and relevant hadronic matrix elements
Then obtain f NQ in terms of f N
q through to MN = 〈N|θµµ|N〉, w/
θµµ = (1− γm(αs))
∑q=u,d,s
mq qq +∑
Q=c,b,t
mQQQ
+β(αs)
2αsG2
Integrate out Q = t , b, c and obtain f Nc from f N
q , q = u, d , s, etc.
Will be done to O(α3s) (Hill et al ’15), but at LO find
f NQ ≡
〈N|mQQQ|N〉MN
=2
27
1−∑
q=u,d,s
f Nq
+ O(αs, α
2s
Λ2QCD
4m2Q
)since 4πβ(αs) = −β0α
2s + O(α3
s) and β0 = 11− 23 Nq − 2
3 NQ
For f Nq , q = u, d , s, use lattice QCD and Feynman-Hellman theorem
f Nq MN = 〈N|mq qq|N〉 = mq
∂MN
∂mq
∣∣∣∣mΦ
q
Laurent Lellouch KEK-PH, , 13-16 February 2017
What is lattice QCD (LQCD)?To describe ordinary matter, QCD requires ≥ 104 numbers at every point of spacetime→∞ number of numbers in our continuous spacetime→ must temporarily “simplify” the theory to be able to calculate (regularization)⇒ Lattice gauge theory −→ mathematically sound definition of NP QCD:
UV (& IR) cutoff→ well defined path integral inEuclidean spacetime:
〈O〉 =
∫DUDψDψ e−SG−
∫ψD[M]ψ O[U, ψ, ψ]
=
∫DU e−SG det(D[M]) O[U]Wick
DUe−SG det(D[M]) ≥ 0 & finite # of dofs→ evaluate numerically using stochastic methods
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
-6�?
6
?
T
-�L
?6a
Uµ(x) = eiagAµ(x) ψ(x)
LQCD is QCD but only when Nf ≥ 2 + 1, mq → mphysq , a→ 0, L→∞
HUGE conceptual and numerical challenge (integrate over ∼ 109 real variables)
⇒ very few calculations control all necessary limitsLaurent Lellouch KEK-PH, , 13-16 February 2017
Strategy of calculation
Objective:
Determine slope of MN wrt mq , q = u, d , s, at physical point
Method:
Perform many high-statistics simulations with various mq around physical values, variousa <∼ 0.1 fm and various L >∼ 6 fm
For each compute Mπ (→ mud ), Mηs =√
2M2K −M2
π (→ ms), MDs (→ mc ) and MN
(→ ΛQCD)
Study dependence of mq , q = ud , s, c and MN on Mπ , Mηs , MDs , a and L
For each simulation determine a, mΦq ’s such that Mπ , . . . take their physical value in a→ 0
and L→∞ limit
Compute, at physical point
f Nq =
∑P=π,ηs
∂ ln M2P
∂ ln mq
∂ ln MN
∂ ln M2P
Laurent Lellouch KEK-PH, , 13-16 February 2017
Lattice details
Nf = 1 + 1 + 1 + 13HEX clover-improved Wilson fermions on tree-level improved Symanzikgluons33 ensembles w/ total ∼ 169000 trajectories∼ 500 measurements per configuration4 a ∈ [0.064,0.102] fm;Mπ ∈ [195,450]MeV w/ LMπ > 4
Improvements over BMWc, PRL ’16
3 Charm in sea3 >∼ × 100 in statistics3 >∼ × 2 lever arm in ms
3 Like PRL ’16 FH in terms of quark and not meson masses7 No physical mud , but small enough and know MN from experiment
Laurent Lellouch KEK-PH, , 13-16 February 2017
M2π ∼ mud dependence of MN (preliminary)
1002 2002 3002
Mπ2 [MeV2]
900
950
1000
1050
1100
1150M
N [M
eV]
β=3.2β=3.2 QCD+QEDβ=3.3β=3.4β=3.5
Laurent Lellouch KEK-PH, , 13-16 February 2017
Mηs ∼ ms dependence of MN (preliminary)
3002 4002 5002 6002 7002 8002
Mηs
2 [MeV2]
880
900
920
940
960M
N [M
eV]
β=3.2β=3.2 QCD+QEDβ=3.3β=3.4β=3.5
Laurent Lellouch KEK-PH, , 13-16 February 2017
Chain rule conversion matrix (preliminary)
Have:
∂ ln MN
∂ ln M2P
w/ P = π, ηs
Want:
∂ ln MN
∂ ln mq
w/ q = u, d , s 3002 4002 5002 6002 7002 8002
Mη
2 [MeV2 ]
β=3.2β=3.3β=3.4β=3.5
1002 2002 3002
Mπ
2 [MeV2 ]
0
0.5
1
1.5
ms/m
sφ
0
0.1
0.2
mud
/msφ
s
∂ ln M2π
∂ ln mud
∂ ln M2ηs
∂ ln mud∂ ln M2
π∂ ln ms
∂ ln M2ηs
∂ ln ms
=
(0.94(1)(1) 0.002(0)(1)0.06(2)(3) 1.02(0)(2)
)
Laurent Lellouch KEK-PH, , 13-16 February 2017
Systematic error assessment (preliminary)Estimated using extended frequentist approach (BMWc, Science ’08, Science ’15)
Excited state contamination: 4 time intervals for correlations functions
Mass interpolation/extrapolation errors
Mπ ≤ 330/360/420 MeVdifferent Mπ/ηs dependences (polynomials, Padés, χPT)
continuum extrapolation: O(αsa) vs O(a2)
⇒ 672 analyses which differ by higher order effects
20 30 40 50σudN [MeV]
0
0.2
0.4
0.6
0.8
1
rela
tive
wei
ght
AIC weightQ weightflat weight
40 50 60 70σsN [MeV]
0
0.2
0.4
0.6
0.8
1
rela
tive
wei
ght
AIC weightQ weightflat weight
Laurent Lellouch KEK-PH, , 13-16 February 2017
Preliminary results
Direct results
f Nud = 0.0430(19)(32) [8.6%] f N
s = 0.0564(38)(35) [9.2%]
Using SU(2) isospin (BMWc, PRL116) w/ ∆QCDMN = 2.52(17)(24) MeV (BMWc,Science 347) & mu/md = 0.485(11)(16) (BMWc, PRL117)
f pu = 0.0153(6)(11) [8.1%] f p
d = 0.0264(13)(21) [9.4%]
f nu = 0.0128(6)(11) [9.7%] f n
d = 0.0316(13)(21) [7.9%]
Using f Nud,s & HQ expansion up to O(α4
s ,Λ2QCD/m
2c ) corrections (Hill et al ’15)
f Nc = 0.0730(5)(5)(??) [1.0 + ??%] f N
b = 0.0700(4)(4)(??) [0.9 + ??%]
f Nt = 0.0678(3)(3)(??) [0.7 + ??%]
Laurent Lellouch KEK-PH, , 13-16 February 2017
Low-energy effective h-N coupling (preliminary)
1 100 10000mq [MeV]
0
2
4
6
8
f fp [%]
u
d
s
cb t
ffN is q contribution to effective coupling of Higgs to nucleon in units of MN
⇒ fraction of MN coming from q contribution to coupling of N to Higgs vev
HQ expansion⇒ Q = c, b, t contributions mainly through their impact on the running of αs
fN = 0.310(3)(3)(??) [1.4 + ??%]
Laurent Lellouch KEK-PH, , 13-16 February 2017
Conclusion
Scalar quark contents of p & n have been computed with full control overall sources of uncertainties
Important for: DM searches; coherent LFV µ→ e conversion in nuclei;describing low-energy coupling of N to the Higgs; understanding MN ; πN and KNscattering, etc.
f Nq , q = u,d , s & N = n,p are now known to better than 10%
f NQ , Q = c,b, t and fN to even better precision
Correlations between the various quantities will be given
Hadronic ME are no longer the dominant source of uncertainty in DMdirect detection rate predictions . . .
. . . or in the determination of WIMP couplings from possible DM signals
Laurent Lellouch KEK-PH, , 13-16 February 2017
BACKUP
Laurent Lellouch KEK-PH, , 13-16 February 2017
Comparison
GLS 91 _Pavan 02 _Alarcon et al 12 _Shanahan et al 12 _Alvarez et al 13, FH _Lutz et al 14, FH _Ren et al 14, FH _Hoferichter et al 15 _JLQCD 08, FH _Bali et al 11, FH _ETM 16, ME _Bali et al 16, ME _BMWc 11, FH _QCDSF 11,FH _
BMWc 16, FH _
Yang et al 15, ME _
New dataset, FH_
0.02
0.02
0.04
0.04
0.06
0.06
0.08
0.08fN
ud
Pheno.Nf=2Nf=2+1Nf=2+1+1
MILC 09, FH _BMWc 11, FH _QCDSF 11, FH _Ohki et al 13, ME _Junnarkar et al 13, FH _Gong et al 13, ME _Yang et al 15, ME _
BMWc 16, FH _
GLS 91 _Pavan 02 _
Shanahan et al 12 _Lutz et al 14 _Ren et al 14 _An et al 14 _Alarcon et al 14 _JLQCD 10, FH _Bali et al 11, ME _ETM 16, ME _Bali et al 16, ME _
New dataset, FH _0
0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5fN
s
Pheno.Nf=2Nf=2+1Nf=2+1+1
Laurent Lellouch KEK-PH, , 13-16 February 2017
Finite-volume effects
1/10 1/6 1/4 1/3L-1 [fm-1]
940
960
980
1000
MN[M
eV]
β=3.2β=3.2 QCD+QEDβ=3.3β=3.4β=3.5
Fit away leading effects MX (L)−MXMX
= cM1/2π L−3/2e−LMπ
Compabtible w/ χPT expectation (Colangelo et al ’10)
Laurent Lellouch KEK-PH, , 13-16 February 2017
Kolmogorov-Smirnov test and ground state extraction
Selection of fit-time range is crucial and delicate to isolate N ground state incorrelation functions
Consider cumulative distributions ofthe fit qualities over 33 ensembles fordifferent tmin and hadrons
Fit quality should be uniformlydistributed
Apply Kolmogorov-Smirnov analysis totest measured distributions
Keep tmin 3 distribution compatible w/uniform distribution w/ prob. > 30%
0 0.2 0.4 0.6 0.8 1fit quality
0
0.2
0.4
0.6
0.8
1
cdf(M
p)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
tref=1 fm p=0.09tref=1.3 fm p=0.55
Laurent Lellouch KEK-PH, , 13-16 February 2017
Systematic error decomposition
3 All analyses in goodagreement
3 Reasonable fitquality0.03 < Q < 0.65
50 60 70σsN [MeV]
t0=-1.25 Q=0.20t0=-1.3 Q=0.23
t0=-1.35 Q=0.34t0=-1.4 Q=0.24
no a Q=0.23αa q Q=0.23a2 q Q=0.23
αa N Q=0.0.27αa N αa q Q=0.0.27
a2 N Q=0.27a2 N a2 q Q=0.27
Mπ<330 MeV Q=0.30
Mπ<360 MeV Q=0.26
Mπ<420 MeV Q=0.20
Mπ
4 Q=0.22M
π
3 Q=0.29
Mχ
2 Pade Q=0.24M
χ
2 Taylor Q=0.26
LO mq Q=0.25NLO mq Q=0.25
TOTAL Q=0.25
30 40 50σudN [MeV]
Laurent Lellouch KEK-PH, , 13-16 February 2017
Example continuum extrapolation of MN (preliminary)
0.0502 0.1002
a[fm]
920
930
940
950
960
970
980
MN[M
eV]
Laurent Lellouch KEK-PH, , 13-16 February 2017
New method for obtaining f p/nu/d
[BMWc, PRL 116 (2016)]
Input: f Nud and ∆QCDMN = Mn − Mp (from BMWc, Science ’15)
SU(2) relations w/ δm = md − mu
H = Hiso + Hδm , Hδm =δm2
∫d3x (dd − uu)
∆QCDMN = δm〈p|uu − dd |p〉
lead to, w/ r = mu/md ,
f p/nu =
(r
1 + r
)f Nud ±
12
(r
1− r
)∆QCDMN
MN
f p/nd =
(1
1 + r
)f Nud ∓
12
(1
1− r
)∆QCDMN
MN
Huge improvement on usual SU(3)-flavor approach
systematic:(
ms −mud
ΛQCD
)2≈ 10% −→
(md −mu
ΛQCD
)2≈ 0.01% .
Laurent Lellouch KEK-PH, , 13-16 February 2017
σ-terms from LQCD: matrix element (ME) methodσπN = mud〈N|uu + dd |N〉 σsN = ms〈N|ss|N〉
Extract directly from time-dependence of 3-pt fns:
N(ti) N(tf)N(tf)
qΓq(t)
N(ti) N(tf)
(t−ti ),(tf−t)→∞−→ 〈N(~0)|qΓq|N(~0)〉
3 Desired matrix element appears at leading order
7 Must compute more noisy 3-pt fn
7 Quark-disconnected contribution difficult, though1/Nc suppressed
7 mq qq renormalization challenging (Wilson fermions)
N(ti) N(tf)
qΓq(t)
Laurent Lellouch KEK-PH, , 13-16 February 2017
σ-terms from LQCD: Feynman-Hellmann (FH) method
Feynman-Hellmann theorem yields:
〈N|mq qq|N〉 = mq∂MN
∂mq
∣∣∣∣mΦ
q
On lattice get MN from time-dependence of 2pt-fn, e.g.:
N(ti) N(tf)(t−ti )→∞−→ 〈0|N|N〉〈N|N|0〉 exp {−MN(tf − ti )}
3 Only simpler and less noisy 2pt-fn is needed
3 No difficult quark-disconnected contributions
3 No difficult renormalization
7 ∂MN/∂mq small for q = [ud ] and even smaller for q = s, c, . . .
Laurent Lellouch KEK-PH, , 13-16 February 2017