Flavorful Electroweak Precision Observables in the Standard

Model Effective Field Theory

Sally Dawson a and Pier Paolo Giardino b

aDepartment of Physics, Brookhaven National Laboratory, Upton, N.Y., 11973, U.S.A.

bInstituto Galego de Fısica de Altas Enerxıas, Universidade de Santiago de Compostela,

15782 Santiago de Compostela, Galicia, Spain

(Dated: January 26, 2022)

Abstract

Electroweak precision observables (EWPO) measured at the W and Z poles provide stringent

limits on possible beyond the Standard Model physics scenarios. In an effective field theory (EFT)

framework, the next-to-leading order QCD and electroweak results for EWPO yield indirect limits

on possible 4-fermion operators that do not contribute to the observables at tree level. Here

we calculate the next-to-leading corrections to EWPO induced by flavor non-universal 4-fermion

interactions and find that the extracted limits on EFT coefficients have a strong dependence on

the flavor structure of the 4-fermion operators.

1

arX

iv:2

201.

0988

7v1

[he

p-ph

] 2

4 Ja

n 20

22

I. INTRODUCTION

Historically, measurements of electroweak precision observables (EWPO) at the W and Z

poles have provided strong bounds on possible beyond the Standard Model (BSM) physics

assumed to occur at some high scale Λ. Similarly, in the Standard Model effective field theory

(SMEFT) approach, comparisons of EWPO with theoretical predictions severely constrain

the allowed values of the coefficients of the effective field theory operators[1–6]. These bounds

rely on precision calculations of the theoretical expectations both in the Standard Model

(SM) and in the SMEFT. SMEFT calculations at next-to-leading order (NLO) QCD can

be automated[7], while a growing number of SMEFT electroweak NLO results exist[1, 8–

20]. In a previous work[1], we computed the O(M2

Z

Λ2 ) NLO QCD and electroweak corrections

to the SMEFT predictions for EWPO at the Z and W poles. At NLO, a dependence on

operators beyond those occurring at tree level in the analysis of EWPO arises and limits can

be placed on the corresponding coefficients. Quark and lepton flavor effects were neglected

in our earlier study and a U(3)5 global flavor symmetry assumed for all operators involving

fermions. Experimental anomalies in B physics, however, suggest that including flavor

dependencies in the SMEFT analyses could be relevant[21, 22]. Furthermore, tree level global

analyses of weak scale processes have a significant dependence on the flavor assumptions that

are made in the fermion sector[23–26]. In this note, we generalize our previous results for

EWPO to include flavor effects from the 4-fermion operators involving at least 2 quarks that

contribute at one-loop level. We present some interesting numerical consequences of allowing

the 4-fermion operators to have an arbitrary flavor structure and compare our results with

those from fits to top quark data at the LHC.

II. BASICS

The SMEFT parameterizes new physics through an expansion in higher dimensional

operators[27],

L = LSM + Σ∞k=5Σnα=1

Ckα

Λk−4Okα , (1)

where the SU(3) × SU(2)L × U(1)Y invariant operators, Okα, are constructed from SM

fields and all of the effects of the BSM physics reside in the coefficient functions, Ckα. We

use the Warsaw basis [28, 29], assume all coefficients are real, include only dimension-6

2

operators and do not consider CP violation. Observables are computed consistently to

O(M2

Z

Λ2 ), corresponding to only one operator insertion at the amplitude level, including all

NLO QCD and electroweak contributions, as described in Ref. [1]. At lowest order (LO),

only the operators Oll,OφWB,OφD,Oφe,Oφu,Oφd,O(3)φq ,O

(1)φq ,O

(3)φl , and O(1)

φl contribute. Ref.

[1] also included the contributions of the operators

Oed ,Oee ,Oeu ,Olu ,Old ,Ole ,O(1)lq ,O(3)

lq ,OφB ,OφW ,O�,

Oqe ,OuB ,OuW ,OW ,O(1)qd ,O

(3)qq ,O(1)

qq ,O(1)qu ,O

(1)ud ,Ouu ,Odd , (2)

that first arise at NLO, where the coefficients of the 2-quark and 4-quark operators were

assumed to be flavor independent. All renormalization group mixing of the operators was

included, as required to obtain a finite NLO result. Contrary to Ref. [1], here we include the

effects of arbitrary flavor interactions between quarks in the 4-fermion operators. In doing

our computation, we set the SM CKM matrix to be the unit matrix and we assumed that

the 2-fermion operators that appear at lowest order are diagonal, but not proportional to

unity. For example1

Oφu,[11] 6= Oφu,[22] 6= Oφu,[33] Oφu,[12] = Oφu,[13] = Oφu,[23] = Oφu,[13] = 0 . (3)

It is worth noting that setting the SM CKM matrix to 1 is enough to ensure that at LO

only the diagonal terms of the 2-fermion operators contribute to all Z-pole observables. At

NLO, the analysis becomes more complicated. However, since we are interested only in the

contributions induced by the 4-fermion operators, it is straightforward to observe that our

choice of taking the SM CKM matrix to be unity affects our results mostly through the

renormalization group (RGE) mixing of operators. The only observable where this is not

true is the W width where extra terms proportional to the off-diagonal terms of the SM

CKM matrix are not included. For these reasons, at the level we are working, we expect the

effects on our results for the 4−fermion interactions from our choice of taking the SM CKM

matrix to be unity to be small.

1 After renormalization, we drop the generation indices on the 2-fermion operators.Therefore, the results

presented here differ from those of Ref. [1] only for the 4-quark and 2-quark, 2-lepton operators listed in

Table I.

3

2-quark 2-lepton operators

O(1)lq,[tp] (lLγµlL)(qL,tγ

µqL,p) O(3)lq,[tp] (lLγµσ

AlL)(qL,tγµσAqL,p)

Oeu,[tp] (eRγµeR)(uR,tγµuR,p) Oed,[tp] (eRγµeR)(dR,tγ

µdR,p)

Olu,[tp] (lLγµlL)(uR,tγµuR,p) Old,[tp] (lLγµlL)(dR,tγ

µdR,p)

Oqe,[rs] (qL,rγµqL,s)(eRγµeR)

4-quark operators

O(1)qq,[rstp] (qL,rγµqL,s)(qL,tγ

µqL,p) O(3)qq,[rstp] (qL,rγµσ

AqL,s)(qL,tγµσAqL,p)

Ouu,[rstp] (uR,rγµuR,s)(uR,tγµuR,p) Odd,[rstp] (dR,rγµdR,s)(dR,tγ

µdR,p)

O(1)ud,[rstp] (uR,rγµuR,s)(dR,tγ

µdR,p) O(1)qu,[rstp] (qL,rγµqL,s)(uR,tγ

µuR,p)

O(1)qd,[rstp] (qL,rγµqL,s)(dR,tγ

µdR,p)

TABLE I: Dimension-6 4 -fermion operators contributing to Z and W pole observables at NLO

QCD and electroweak order [1] where [r, s, t, p] = 1, 2, 3 are quark generation indices and σA are

the Pauli matrices.

Defining the fermion fields as,

qL,r =

uL,r

dL,r

, uR,r, dR,r, lL,r =

νL,r

eL,r

, eR,r , (4)

where r = 1, 2, 3 is a generation index, we considered the operators in Table I, where we

drop the lepton flavor indices.

III. RESULTS

The observables we consider are:

MW ,ΓW ,ΓZ , σh, Rl, Al,FB, Rb, Rc, AFB,b, AFB,c, Ab, Ac, Al . (5)

4

Z

u

u

1

FIG. 1: Sample diagram containing 4− fermion operators contributing to Z → uu at NLO in the

SMEFT. The fermions in the loop can be any quark or lepton (heavy or light). The red circle

represents insertions of the operators of Table I.

The SM results for these observables are quite precisely known and we use the experimental

and theoretical SM results shown in Table III of Ref. [1]. The NLO SMEFT results for the

observables of Eq. 5 contain one-loop contributions from the dimension-6 operators of Table

I and the full electroweak and QCD NLO amplitudes assuming that the flavor interactions

are independent of fermion generation are in the supplemental material of Ref. [1]. Here

we focus on the effects of the 4-fermion operators for on-shell 2-body Z and W decays such

as those shown in Fig. 1 and allow for an arbitrary flavor dependence in the 4−fermion

operators. When the internal fermions are top quarks, contributions that are enhanced by

factors of M2t /M

2Z arise. Such contributions contribute to Z → bb generically through the

coefficients, Cα,[3333], and to Z → fif i (where fi is a light fermion) through the coefficients,

Cα,[33ii], etc. We note that not all combinations of generation indices arise in the NLO

calculation of the EWPO. For example, the operator O(1)qq occurs with i, i′ = 1, 2, (where

i 6= i′)

C(1)qq,[3333], C

(1)qq,[33ii] = C

(1)qq,[ii33], C

(1)qq,[3ii3] = C

(1)qq,[i33i], C

(1)qq,[iii′i′], C

(1)qq,[ii′i′i] C

(1)qq,[iiii] . (6)

In our calculation we never encounter operators with more than 2 different flavor indices,

due to our choice of flavor structure.

The SMEFT predictions for the observables are,

OSMEFT,LOi = OSM,LO

i + δOLOi (Cj)

OSMEFT,NLOi = OSM,NLO

i + δONLOi (Cj) . (7)

We present numerical results for the observables of Eq. 5 in the supplemental material

attached to this note. In the limit where the Cα,[rstp] are independent of the generation

5

indices, our previous result is recovered.

We perform a χ2 fit to the EWPO data. As an example, if only the 3rd generation quarks

contribute to the 4- fermion operators, the NLO contribution to the χ2 is2

∆χ2[3333] = −0.45893C

(1)ud,[3333] + 1.0712C

(1)qu,[3333] − 0.16008C

(3)qq,[3333] − 1.7914C

(1)qq,[3333]

+0.37024C(1)qd,[3333] + 0.019645Cdd,[3333] + ~XTM[3333]

~X , (8)

with ~XT = (C(1)ud,[3333], C

(1)qu,[3333], C

(3)qq,[3333], C

(1)qq,[3333], C

(1)qd,[3333], Cdd,[3333]) and

M[3333] =

0.039703 −0.38683 +0.057807 +0.6469 −0.060256 −0.0033991

1.1917 −0.35618 −3.9858 +0.28411 +0.016559

0.026614 +0.59564 −0.042458 −0.0024745

3.3328 −0.47512 −0.027691

0.022951 0.0025794

7.2752× 10−5

. (9)

It is apparent that the largest sensitivity in this case is to C(1)qq,[3333] and to C

(1)qu,[3333].

We show the 95% confidence level single parameter limits on the 4−fermion operators of

Table I with various flavor assumptions in Table II. All of our coefficients are evaluated at

a scale MZ . Column 2 is the result of Ref. [1] where there is no flavor dependence in the

4−fermion coefficients,

Column 2 of Tab. II: Cα,[3333] = Cα,[33ii] = Cα,[ii33] = Cα,[3ii3] = Cα,[i33i]

= Cα,[iiii] = Cα,[iii′i′] = Cα,[ii′i′i] . (10)

It is of interest to consider the scenario where the 3rd generation 4−fermion operators are

different from those of generations i = 1, 2. Column 3 assumes that the 4−fermion operators

are only generated for the 3rd generation quarks and can be obtained from Eqs. 8 and 9,

Column 3 of Tab. II: Cα,[3333] 6= 0, Cα,[33ii] = Cα,[ii33] = Cα,[i33i] = Cα,[3ii3] = 0

Cα,[iii′i′] = Cα,[ii′i′i] = Cα,[iiii] = 0 . (11)

2 ∆χ2[3333] must be added to the full SMEFT result including all other operators.

6

-100

-50

0

50

100

C/Λ

2 [TeV

-2]

Flavor independent 4fOnly 3rd genAt least 3rd genOnly 1st and 2nd gen

95% CL limits from NLO EWPO on 4-fermion operators

<latexit sha1_base64="ofOGvVRY0SnmOEN2koeZ7kCng9Q=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkIiRT0WevFYwX5AG8tmu2mXbjbp7kYoIX/DiwdFvPpnvPlv3LY5aOuDgcd7M8zM82POlHacb6uwsbm1vVPcLe3tHxwelY9P2ipKJKEtEvFIdn2sKGeCtjTTnHZjSXHoc9rxJ42533miUrFIPOhZTL0QjwQLGMHaSP3GIJ1Os8e06l5mg3LFsZ0F0Dpxc1KBHM1B+as/jEgSUqEJx0r1XCfWXoqlZoTTrNRPFI0xmeAR7RkqcEiVly5uztCFUYYoiKQpodFC/T2R4lCpWeibzhDrsVr15uJ/Xi/Rwa2XMhEnmgqyXBQkHOkIzQNAQyYp0XxmCCaSmVsRGWOJiTYxlUwI7urL66R9ZbvXtntfq9TtPI4inME5VMGFG6jDHTShBQRieIZXeLMS68V6tz6WrQUrnzmFP7A+fwBri5E4</latexit>

C(1)qq

<latexit sha1_base64="F3kUcVdZWL7i/olrnNBcPF1GMJ0=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkKioh4LvXisYD+gjWWz3bRLN5t0dyOUkL/hxYMiXv0z3vw3btsctPXBwOO9GWbm+TFnSjvOt1VYW9/Y3Cpul3Z29/YPyodHLRUlktAmiXgkOz5WlDNBm5ppTjuxpDj0OW374/rMbz9RqVgkHvQ0pl6Ih4IFjGBtpF69n04m2WNavTzP+uWKYztzoFXi5qQCORr98ldvEJEkpEITjpXquk6svRRLzQinWamXKBpjMsZD2jVU4JAqL53fnKEzowxQEElTQqO5+nsixaFS09A3nSHWI7XszcT/vG6ig1svZSJONBVksShIONIRmgWABkxSovnUEEwkM7ciMsISE21iKpkQ3OWXV0nrwnavbff+qlKz8ziKcAKnUAUXbqAGd9CAJhCI4Rle4c1KrBfr3fpYtBasfOYY/sD6/AFul5E6</latexit>

C(3)qq

<latexit sha1_base64="V90e9U3+pVq2ue3S5I1oYnd/dGg=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2FXRD0GcvEYwTwgWcPsZDYZMju7zkMIy/6GFw+KePVnvPk3TpI9aGJBQ1HVTXdXkHCmtOt+O4W19Y3NreJ2aWd3b/+gfHjUVrGRhLZIzGPZDbCinAna0kxz2k0kxVHAaSeYNGZ+54lKxWJxr6cJ9SM8EixkBGsr9RuD9NFkD2nVO88G5Ypbc+dAq8TLSQVyNAflr/4wJiaiQhOOlep5bqL9FEvNCKdZqW8UTTCZ4BHtWSpwRJWfzm/O0JlVhiiMpS2h0Vz9PZHiSKlpFNjOCOuxWvZm4n9ez+jwxk+ZSIymgiwWhYYjHaNZAGjIJCWaTy3BRDJ7KyJjLDHRNqaSDcFbfnmVtC9q3lXNu7us1Gt5HEU4gVOoggfXUIdbaEILCCTwDK/w5hjnxXl3PhatBSefOYY/cD5/AHG3kTw=</latexit>

C(1)qu

<latexit sha1_base64="AAKGj8U1hQN+GgoftRcFlY/JtGc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkIioh4LvXisYD+gjWWz2bRLN5u4uxFKyN/w4kERr/4Zb/4bt20O2vpg4PHeDDPz/IQzpR3n2yqtrW9sbpW3Kzu7e/sH1cOjjopTSWibxDyWPR8rypmgbc00p71EUhz5nHb9SXPmd5+oVCwW93qaUC/CI8FCRrA20qA5zB6D/CGru+f5sFpzbGcOtErcgtSgQGtY/RoEMUkjKjThWKm+6yTay7DUjHCaVwapogkmEzyifUMFjqjysvnNOTozSoDCWJoSGs3V3xMZjpSaRr7pjLAeq2VvJv7n9VMd3ngZE0mqqSCLRWHKkY7RLAAUMEmJ5lNDMJHM3IrIGEtMtImpYkJwl19eJZ0L272y3bvLWsMu4ijDCZxCHVy4hgbcQgvaQCCBZ3iFNyu1Xqx362PRWrKKmWP4A+vzB1d8kSs=</latexit>

C(1)qd

<latexit sha1_base64="n/FqpHlM6cBjfhXFTY74FCL5L2Q=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8Lbsi6jGQi8cI5gHJEmYns8mY2ZllHkJY8g9ePCji1f/x5t84SfagiQUNRVU33V1xxpk2QfDtra1vbG5tl3bKu3v7B4eVo+OWllYR2iSSS9WJsaacCdo0zHDayRTFacxpOx7XZ377iSrNpHgwk4xGKR4KljCCjZNa9X5u7bRfqQZ+MAdaJWFBqlCg0a989QaS2JQKQzjWuhsGmYlyrAwjnE7LPatphskYD2nXUYFTqqN8fu0UnTtlgBKpXAmD5urviRynWk/S2HWm2Iz0sjcT//O61iS3Uc5EZg0VZLEosRwZiWavowFTlBg+cQQTxdytiIywwsS4gMouhHD55VXSuvTDaz+8v6rW/CKOEpzCGVxACDdQgztoQBMIPMIzvMKbJ70X7937WLSuecXMCfyB9/kDuryPLA==</latexit>

Cuu

<latexit sha1_base64="v1i7v+JPDm1alDsUkZqI33D8EL8=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0hEqsdCLx4r2A9oQ9lsNu3azW7Y3Qgl9D948aCIV/+PN/+N2zYHbX0w8Hhvhpl5YcqZNp737ZQ2Nre2d8q7lb39g8Oj6vFJR8tMEdomkkvVC7GmnAnaNsxw2ksVxUnIaTecNOd+94kqzaR4MNOUBgkeCRYzgo2VOs1hHkWzYbXmud4CaJ34BalBgdaw+jWIJMkSKgzhWOu+76UmyLEyjHA6qwwyTVNMJnhE+5YKnFAd5ItrZ+jCKhGKpbIlDFqovydynGg9TULbmWAz1qveXPzP62cmvg1yJtLMUEGWi+KMIyPR/HUUMUWJ4VNLMFHM3orIGCtMjA2oYkPwV19eJ50r16+7/v11reEWcZThDM7hEny4gQbcQQvaQOARnuEV3hzpvDjvzseyteQUM6fwB87nD4cBjwo=</latexit>

Cdd

<latexit sha1_base64="8PhxwZE6HTfssnIjq4Re2v7VG48=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkIioh4LvXisYGuhjWWz2bRLN5uwH0IJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZemHGmtOd9O6W19Y3NrfJ2ZWd3b/+genjUUamRhLZJylPZDbGinAna1kxz2s0kxUnI6UM4bs78hycqFUvFvZ5kNEjwULCYEayt1G8OchNNH/O6fz4dVGue682BVolfkBoUaA2qX/0oJSahQhOOler5XqaDHEvNCKfTSt8ommEyxkPas1TghKogn988RWdWiVCcSltCo7n6eyLHiVKTJLSdCdYjtezNxP+8ntHxTZAzkRlNBVksig1HOkWzAFDEJCWaTyzBRDJ7KyIjLDHRNqaKDcFffnmVdC5c/8r17y5rDbeIowwncAp18OEaGnALLWgDgQye4RXeHOO8OO/Ox6K15BQzx/AHzucPXayRLw==</latexit>

C(1)ud

<latexit sha1_base64="pGRL600aDrDFLzLlDv+Rrzaa/aM=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LBahXkIioh4LvXisYD+gjWWz3bRLN5u4uxFKyN/w4kERr/4Zb/4bt20O2vpg4PHeDDPzgoQzpV332yqtrW9sbpW3Kzu7e/sH9uFRW8WpJLRFYh7LboAV5UzQlmaa024iKY4CTjvBpDHzO09UKhaLez1NqB/hkWAhI1gbqd8YZPwxf8hq3nk+sKuu486BVolXkCoUaA7sr/4wJmlEhSYcK9Xz3ET7GZaaEU7zSj9VNMFkgke0Z6jAEVV+Nr85R2dGGaIwlqaERnP190SGI6WmUWA6I6zHatmbif95vVSHN37GRJJqKshiUZhypGM0CwANmaRE86khmEhmbkVkjCUm2sRUMSF4yy+vkvaF41053t1lte4UcZThBE6hBh5cQx1uoQktIJDAM7zCm5VaL9a79bFoLVnFzDH8gfX5A2PPkTM=</latexit>

C(1)lq

<latexit sha1_base64="REikaDp1+JphIdl0yR8Pk7RkT0w=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkJiRT0WevFYwX5AG8tmu2mXbjZxdyOUkL/hxYMiXv0z3vw3btsctPXBwOO9GWbm+TFnSjvOt1VYW9/Y3Cpul3Z29/YPyodHbRUlktAWiXgkuz5WlDNBW5ppTruxpDj0Oe34k8bM7zxRqVgk7vU0pl6IR4IFjGBtpH5jkPLH7CGt1s6zQbni2M4caJW4OalAjuag/NUfRiQJqdCEY6V6rhNrL8VSM8JpVuonisaYTPCI9gwVOKTKS+c3Z+jMKEMURNKU0Giu/p5IcajUNPRNZ4j1WC17M/E/r5fo4MZLmYgTTQVZLAoSjnSEZgGgIZOUaD41BBPJzK2IjLHERJuYSiYEd/nlVdK+sN0r2727rNTtPI4inMApVMGFa6jDLTShBQRieIZXeLMS68V6tz4WrQUrnzmGP7A+fwBm25E1</latexit>

C(3)lq

<latexit sha1_base64="/ro791yJlAg1AeE8d5K0hAmLL6Q=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8Lbsi6jGQi8cI5gHJEmYns8mY2ZllHkJY8g9ePCji1f/x5t84SfagiQUNRVU33V1xxpk2QfDtra1vbG5tl3bKu3v7B4eVo+OWllYR2iSSS9WJsaacCdo0zHDayRTFacxpOx7XZ377iSrNpHgwk4xGKR4KljCCjZNa9X7O7bRfqQZ+MAdaJWFBqlCg0a989QaS2JQKQzjWuhsGmYlyrAwjnE7LPatphskYD2nXUYFTqqN8fu0UnTtlgBKpXAmD5urviRynWk/S2HWm2Iz0sjcT//O61iS3Uc5EZg0VZLEosRwZiWavowFTlBg+cQQTxdytiIywwsS4gMouhHD55VXSuvTDaz+8v6rW/CKOEpzCGVxACDdQgztoQBMIPMIzvMKbJ70X7937WLSuecXMCfyB9/kDrQaPIw==</latexit>

Clu<latexit sha1_base64="To4o+g3O9jIfhv+Oi9XLRyzJ2hc=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0hEqsdCLx4r2A9oQ9lsNu3azW7Y3Qgl9D948aCIV/+PN/+N2zYHbX0w8Hhvhpl5YcqZNp737ZQ2Nre2d8q7lb39g8Oj6vFJR8tMEdomkkvVC7GmnAnaNsxw2ksVxUnIaTecNOd+94kqzaR4MNOUBgkeCRYzgo2VOs1hzqPZsFrzXG8BtE78gtSgQGtY/RpEkmQJFYZwrHXf91IT5FgZRjidVQaZpikmEzyifUsFTqgO8sW1M3RhlQjFUtkSBi3U3xM5TrSeJqHtTLAZ61VvLv7n9TMT3wY5E2lmqCDLRXHGkZFo/jqKmKLE8KklmChmb0VkjBUmxgZUsSH4qy+vk86V69dd//661nCLOMpwBudwCT7cQAPuoAVtIPAIz/AKb450Xpx352PZWnKKmVP4A+fzB5MxjxI=</latexit>

Cld

<latexit sha1_base64="EWqxTd9gFmdOla6lii3C0f4yFms=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6rHQi8cK9gPaUDbbSbt2sxt3N0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemHCmjed9O4W19Y3NreJ2aWd3b/+gfHjU0jJVFJtUcqk6IdHImcCmYYZjJ1FI4pBjOxzXZ377CZVmUtybSYJBTIaCRYwSY6VWvZ894rRfrnhVbw53lfg5qUCORr/81RtImsYoDOVE667vJSbIiDKMcpyWeqnGhNAxGWLXUkFi1EE2v3bqnlll4EZS2RLGnau/JzISaz2JQ9sZEzPSy95M/M/rpia6CTImktSgoItFUcpdI93Z6+6AKaSGTywhVDF7q0tHRBFqbEAlG4K//PIqaV1U/auqf3dZqVXzOIpwAqdwDj5cQw1uoQFNoPAAz/AKb450Xpx352PRWnDymWP4A+fzB5xUjxg=</latexit>

Cqe

<latexit sha1_base64="XBzMJoVrAhG7iZolPHX+ei0eK1c=">AAAB7XicbVBNS8NAEJ34WetX1aOXYBE8lUREPRZ68VjBfkAbymYzaddudsPuRiih/8GLB0W8+n+8+W/ctjlo64OBx3szzMwLU8608bxvZ219Y3Nru7RT3t3bPzisHB23tcwUxRaVXKpuSDRyJrBlmOHYTRWSJOTYCceNmd95QqWZFA9mkmKQkKFgMaPEWKndGOQYTQeVqlfz5nBXiV+QKhRoDipf/UjSLEFhKCda93wvNUFOlGGU47TczzSmhI7JEHuWCpKgDvL5tVP33CqRG0tlSxh3rv6eyEmi9SQJbWdCzEgvezPxP6+Xmfg2yJlIM4OCLhbFGXeNdGevuxFTSA2fWEKoYvZWl46IItTYgMo2BH/55VXSvqz51zX//qparxVxlOAUzuACfLiBOtxBE1pA4RGe4RXeHOm8OO/Ox6J1zSlmTuAPnM8fiIePCw==</latexit>

Ced

<latexit sha1_base64="IwUmzNAXRjQZWhaIFy3nky049i4=">AAAB7XicbVBNS8NAEJ34WetX1aOXYBE8hUREPRZ68VjBfkAbymY7addudsPuRiih/8GLB0W8+n+8+W/ctjlo64OBx3szzMyLUs608f1vZ219Y3Nru7RT3t3bPzisHB23tMwUxSaVXKpORDRyJrBpmOHYSRWSJOLYjsb1md9+QqWZFA9mkmKYkKFgMaPEWKlV7+eYTfuVqu/5c7irJChIFQo0+pWv3kDSLEFhKCdadwM/NWFOlGGU47TcyzSmhI7JELuWCpKgDvP5tVP33CoDN5bKljDuXP09kZNE60kS2c6EmJFe9mbif143M/FtmDORZgYFXSyKM+4a6c5edwdMITV8YgmhitlbXToiilBjAyrbEILll1dJ69ILrr3g/qpa84o4SnAKZ3ABAdxADe6gAU2g8AjP8ApvjnRenHfnY9G65hQzJ/AHzucPolyPHA==</latexit>

Ceu

FIG. 2: Comparison of single parameter limits from loop corrections to EWPO with different

flavor assumptions described in the text.

The 4th column assumes that 4−fermions operators where only 1st and 2nd generation quarks

appear are zero and the remaining operators are equal,

Column 4 of Tab. II: Cα,[3333] = Cα,[33ii] = Cα,[ii33] = Cα,[3ii3] = Cα,[i33i] 6= 0

Cα,[iii′i′] = Cα,[ii′i′i] = Cα,[iiii] = 0 . (12)

Finally, the 5th column assumes that there are no contributions to 4−fermion operators from

3rd generation quarks and that all the coefficients involving 1st and 2nd generation quarks

for each operator are equal,

Column 5 of Tab. II: Cα,[3333] = Cα,[33ii] = Cα,[ii33] = Cα,[3ii3] = Cα,[i33i] = 0 ,

Cα,[iii′i′] = Cα,[ii′i′i] = Cα,[iiii] 6= 0 . (13)

It is obvious that the results are extremely sensitive to the flavor assumptions and that

there are large cancellations between contributions with massive internal top quarks and the

massless fermions. We plot these results in Fig. 2. Figs. 3 and 4 show several 2- parameter

fits with various assumptions about the flavor structure of the 4-fermion operators. Again,

the numerical values of the fits are depend on the generation structure assumed for the

4-fermion operators and strong correlations are observed.

In Table III, we compare the limits from the single parameter EWPO NLO fits to those

obtained from fits to LHC tt data[30–33], using the notation of Ref. [34]. Our NLO fits are

7

-200 -100 0 100 200

Cii/Λ2 [TeV-2]

-5

0

5

C33

/Λ2 [T

eV-2

]

Clq(1)

Clq(3)

FIG. 3: Fit to EWPO with the only non-zero Wilson coefficients being C(1)lq and C

(3)lq . The quark

generation index i = 1, 2.

-20 -10 0 10 20Ciiii/Λ

2 = Cii'i'i /Λ2 = Ciii'i' /Λ2 [TeV-2]

-4

-3

-2

-1

0

1

2

3

C3i

i3/Λ

2 =Ci3

3i/Λ

2 =Cii3

3/Λ2 =C

33ii/Λ

2 =C33

33/Λ

2 [TeV

-2]

Cqq(1)

Cqq(3)

-20 -10 0 10 20 30C3333/Λ

2 [TeV2]

-4

-3

-2

-1

0

1

2

3

C3i

i3/Λ

2 =Ci3

3i/Λ

2 =Cii3

3/Λ2 =C

33ii/Λ

2 [TeV

2 ] Cqq(1)

Cqq(3)

Ciiii/Λ2 = Cii'i'i /Λ2 = Ciii'i' /Λ

2 = 0

FIG. 4: Fit to EWPO with the only non-zero Wilson coefficients C(1)qq and C

(3)qq . The quark

generation index i = 1, 2 and i 6= i′.

only consistent to O( 1Λ2 ), and extending them to O( 1

Λ4 ) would require double insertions of

dimension-6 operators and the inclusion of dimension-8 operators. Comparing the O( 1Λ2 )

EWPO and the tt results, we see that for several operators the EWPO result is comparable

or better than the top quark result. We note, however, that most of the power of the tt fit

is coming at O( 1Λ4 ). It is amusing to note that the EWPO results for C

(1)QQ and C

(1)Qt are still

relevant even when compared to the tt O( 1Λ4 ) results. The impact of the EWPO NLO data

is illustrated graphically in Fig. 5 and suggests that including the NLO EWPO result with

non-universal flavor effects in the global fits could have a significant effect.

8

<latexit sha1_base64="dMw7pkyeaOPiEeE1fzn19nkf2QU=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkoioh4LvXhswX5AG8tms2mXbjZhdyOUkL/hxYMiXv0z3vw3btMctPXBwOO9GWbmeTFnStv2t1Xa2Nza3invVvb2Dw6PqscnPRUlktAuiXgkBx5WlDNBu5ppTgexpDj0OO17s9bC7z9RqVgkHvQ8pm6IJ4IFjGBtpFFrnHb87DGtO5fZuFqzG3YOtE6cgtSgQHtc/Rr5EUlCKjThWKmhY8faTbHUjHCaVUaJojEmMzyhQ0MFDqly0/zmDF0YxUdBJE0JjXL190SKQ6XmoWc6Q6ynatVbiP95w0QHd27KRJxoKshyUZBwpCO0CAD5TFKi+dwQTCQztyIyxRITbWKqmBCc1ZfXSe+q4dw0nM51rVkv4ijDGZxDHRy4hSbcQxu6QCCGZ3iFNyuxXqx362PZWrKKmVP4A+vzByQukQU=</latexit>

C(1)Qd

-40

-20

0

20

40

60

C/Λ

2 [TeV

-2]

EWPO, Λ−2

tt, Λ-2

tt, Λ-4

95% CL limits on 3rd generation 4-fermion operators

<latexit sha1_base64="dMw7pkyeaOPiEeE1fzn19nkf2QU=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkoioh4LvXhswX5AG8tms2mXbjZhdyOUkL/hxYMiXv0z3vw3btMctPXBwOO9GWbmeTFnStv2t1Xa2Nza3invVvb2Dw6PqscnPRUlktAuiXgkBx5WlDNBu5ppTgexpDj0OO17s9bC7z9RqVgkHvQ8pm6IJ4IFjGBtpFFrnHb87DGtO5fZuFqzG3YOtE6cgtSgQHtc/Rr5EUlCKjThWKmhY8faTbHUjHCaVUaJojEmMzyhQ0MFDqly0/zmDF0YxUdBJE0JjXL190SKQ6XmoWc6Q6ynatVbiP95w0QHd27KRJxoKshyUZBwpCO0CAD5TFKi+dwQTCQztyIyxRITbWKqmBCc1ZfXSe+q4dw0nM51rVkv4ijDGZxDHRy4hSbcQxu6QCCGZ3iFNyuxXqx362PZWrKKmVP4A+vzByQukQU=</latexit>

C(1)Qd

<latexit sha1_base64="znxeaz9A/ViN68AGXY+u4VnbKyA=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyURUY+FXjxWsLXQxrLZbNqlm03YnQgl5G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5fiK4Rsf5tkpr6xubW+Xtys7u3v5B9fCoq+NUUdahsYhVzyeaCS5ZBzkK1ksUI5Ev2IM/ac38hyemNI/lPU4T5kVkJHnIKUEjDVrDDIP8Mau75/mwWnMazhz2KnELUoMC7WH1axDENI2YRCqI1n3XSdDLiEJOBcsrg1SzhNAJGbG+oZJETHvZ/ObcPjNKYIexMiXRnqu/JzISaT2NfNMZERzrZW8m/uf1UwxvvIzLJEUm6WJRmAobY3sWgB1wxSiKqSGEKm5utemYKELRxFQxIbjLL6+S7kXDvWq4d5e1Zr2IowwncAp1cOEamnALbegAhQSe4RXerNR6sd6tj0VrySpmjuEPrM8fWlKRKA==</latexit>

C(1)td

<latexit sha1_base64="4GKPhwYgYtKTOitWeUOC+lOwnw4=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2FXRD0GcvGYgHlAsobZyWwyZHZ2mYcQlv0NLx4U8erPePNvnCR70MSChqKqm+6uIOFMadf9dgobm1vbO8Xd0t7+weFR+fiko2IjCW2TmMeyF2BFORO0rZnmtJdIiqOA024wbcz97hOVisXiQc8S6kd4LFjICNZWGjSGactkj2nVu8yG5YpbcxdA68TLSQVyNIflr8EoJiaiQhOOlep7bqL9FEvNCKdZaWAUTTCZ4jHtWypwRJWfLm7O0IVVRiiMpS2h0UL9PZHiSKlZFNjOCOuJWvXm4n9e3+jwzk+ZSIymgiwXhYYjHaN5AGjEJCWazyzBRDJ7KyITLDHRNqaSDcFbfXmddK5q3k3Na11X6tU8jiKcwTlUwYNbqMM9NKENBBJ4hld4c4zz4rw7H8vWgpPPnMIfOJ8/PmmRFg==</latexit>

C(1)Qu

<latexit sha1_base64="1rKyp2CdmPPgaDERMzioqhxLNqo=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkoioh4LvXisYGuhjWWz3bRLN5uwOxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzgkQKg6777RTW1jc2t4rbpZ3dvf2D8uFR28SpZrzFYhnrTkANl0LxFgqUvJNoTqNA8odg3Jj5D09cGxGre5wk3I/oUIlQMIpW6jX6GabTx6zqnU/75Ypbc+cgq8TLSQVyNPvlr94gZmnEFTJJjel6boJ+RjUKJvm01EsNTygb0yHvWqpoxI2fzW+ekjOrDEgYa1sKyVz9PZHRyJhJFNjOiOLILHsz8T+vm2J442dCJSlyxRaLwlQSjMksADIQmjOUE0so08LeStiIasrQxlSyIXjLL6+S9kXNu6p5d5eVejWPowgncApV8OAa6nALTWgBgwSe4RXenNR5cd6dj0VrwclnjuEPnM8fdI2ROQ==</latexit>

C(1)tu

<latexit sha1_base64="nG9H/hgp+benraBf9kShH6dbvZ0=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2FXRHMM5OIxgnlAsobZyWwyZHZ2mYcQlv0NLx4U8erPePNvnCR70MSChqKqm+6uIOFMadf9dgobm1vbO8Xd0t7+weFR+fiko2IjCW2TmMeyF2BFORO0rZnmtJdIiqOA024wbc797hOVisXiQc8S6kd4LFjICNZWGjSHqTbZY1qtX2bDcsWtuQugdeLlpAI5WsPy12AUExNRoQnHSvU9N9F+iqVmhNOsNDCKJphM8Zj2LRU4ospPFzdn6MIqIxTG0pbQaKH+nkhxpNQsCmxnhPVErXpz8T+vb3RY91MmEqOpIMtFoeFIx2geABoxSYnmM0swkczeisgES0y0jalkQ/BWX14nnauad1Pz7q8rjWoeRxHO4Byq4MEtNOAOWtAGAgk8wyu8OcZ5cd6dj2VrwclnTuEPnM8ffzeRQA==</latexit>

C(8)tu

<latexit sha1_base64="M4d1ZKW1yAFTEuXYyaVHj738hQw=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2FXRD0GcvEYwTwgWcPsZDYZMvtwplcIy/6GFw+KePVnvPk3TpI9aGJBQ1HVTXeXF0uh0ba/rcLa+sbmVnG7tLO7t39QPjxq6yhRjLdYJCPV9ajmUoS8hQIl78aK08CTvONNGjO/88SVFlF4j9OYuwEdhcIXjKKR+o1Bio/ZQ1p1zrNBuWLX7DnIKnFyUoEczUH5qz+MWBLwEJmkWvccO0Y3pQoFkzwr9RPNY8omdMR7hoY04NpN5zdn5MwoQ+JHylSIZK7+nkhpoPU08ExnQHGsl72Z+J/XS9C/cVMRxgnykC0W+YkkGJFZAGQoFGcop4ZQpoS5lbAxVZShialkQnCWX14l7Yuac1Vz7i4r9WoeRxFO4BSq4MA11OEWmtACBjE8wyu8WYn1Yr1bH4vWgpXPHMMfWJ8/bmGRNQ==</latexit>

C(1)tq

<latexit sha1_base64="TWxinwuME/kwuGprV7olHC5WMis=">AAAB9XicbVBNS8NAEJ34WetX1aOXxSJUkJKoqMdCLx5bsB/QpmWz3bRLN5u4u1FKyP/w4kERr/4Xb/4bt20O2vpg4PHeDDPzvIgzpW3721pZXVvf2Mxt5bd3dvf2CweHTRXGktAGCXko2x5WlDNBG5ppTtuRpDjwOG154+rUbz1SqVgo7vUkom6Ah4L5jGBtpF61n9Qf0l5Sujx3ztJ+oWiX7RnQMnEyUoQMtX7hqzsISRxQoQnHSnUcO9JugqVmhNM0340VjTAZ4yHtGCpwQJWbzK5O0alRBsgPpSmh0Uz9PZHgQKlJ4JnOAOuRWvSm4n9eJ9b+rZswEcWaCjJf5Mcc6RBNI0ADJinRfGIIJpKZWxEZYYmJNkHlTQjO4svLpHlRdq7LTv2qWCllceTgGE6gBA7cQAXuoAYNICDhGV7hzXqyXqx362PeumJlM0fwB9bnDxpIkYU=</latexit>

C(3,1)Qq

<latexit sha1_base64="AAQPjHpqp3D2AZxKcGJsAIFDbRk=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBahgpRERXss9OKxBfsBbVo22227dLOJuxulhPwPLx4U8ep/8ea/cdvmoK0PBh7vzTAzzws5U9q2v63M2vrG5lZ2O7ezu7d/kD88aqogkoQ2SMAD2fawopwJ2tBMc9oOJcW+x2nLm1RnfuuRSsUCca+nIXV9PBJsyAjWRupV+3H9IenFxauL8nnSzxfskj0HWiVOSgqQotbPf3UHAYl8KjThWKmOY4fajbHUjHCa5LqRoiEmEzyiHUMF9qly4/nVCTozygANA2lKaDRXf0/E2Fdq6num08d6rJa9mfif14n0sOzGTISRpoIsFg0jjnSAZhGgAZOUaD41BBPJzK2IjLHERJugciYEZ/nlVdK8LDk3Jad+XagU0ziycAKnUAQHbqECd1CDBhCQ8Ayv8GY9WS/Wu/WxaM1Y6cwx/IH1+QMk8pGM</latexit>

C(3,8)Qq

<latexit sha1_base64="FOoMwfj19yFA6vakqA5CawiTdz8=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69BItQQUpWRD0WevHYgv2ANi2b7aZdutnE3Y1SQv6HFw+KePW/ePPfuG1z0NYHA4/3ZpiZ50WcKe0431ZubX1jcyu/XdjZ3ds/KB4etVQYS0KbJOSh7HhYUc4EbWqmOe1EkuLA47TtTWozv/1IpWKhuNfTiLoBHgnmM4K1kfq1QdJ4SPtJGV2g83RQLDkVZw57laCMlCBDfVD86g1DEgdUaMKxUl3kRNpNsNSMcJoWerGiESYTPKJdQwUOqHKT+dWpfWaUoe2H0pTQ9lz9PZHgQKlp4JnOAOuxWvZm4n9eN9b+rZswEcWaCrJY5Mfc1qE9i8AeMkmJ5lNDMJHM3GqTMZaYaBNUwYSAll9eJa3LCrquoMZVqVrO4sjDCZxCGRDcQBXuoA5NICDhGV7hzXqyXqx362PRmrOymWP4A+vzBxc4kYM=</latexit>

C(1,1)Qq

<latexit sha1_base64="HFqmPYOgN9neiGupk4USY7Et4qA=">AAAB9XicbVBNS8NAEJ34WetX1aOXxSJUkJKIaI+FXjy2YD+gTctmu2mXbjZxd6OUkP/hxYMiXv0v3vw3btsctPXBwOO9GWbmeRFnStv2t7W2vrG5tZ3bye/u7R8cFo6OWyqMJaFNEvJQdjysKGeCNjXTnHYiSXHgcdr2JrWZ336kUrFQ3OtpRN0AjwTzGcHaSP3aIGk8pP2k5FxWLtJBoWiX7TnQKnEyUoQM9UHhqzcMSRxQoQnHSnUdO9JugqVmhNM034sVjTCZ4BHtGipwQJWbzK9O0blRhsgPpSmh0Vz9PZHgQKlp4JnOAOuxWvZm4n9eN9Z+xU2YiGJNBVks8mOOdIhmEaAhk5RoPjUEE8nMrYiMscREm6DyJgRn+eVV0roqOzdlp3FdrJayOHJwCmdQAgduoQp3UIcmEJDwDK/wZj1ZL9a79bFoXbOymRP4A+vzByHikYo=</latexit>

C(1,8)Qq

<latexit sha1_base64="Z5zPwFXgu1VQDE7xsBnzCNCH2oE=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyURUY+FXjy2YD+gjWWz3bRLN5uwOxFKyN/w4kERr/4Zb/4bt20O2vpg4PHeDDPz/FhwjY7zbRU2Nre2d4q7pb39g8Oj8vFJR0eJoqxNIxGpnk80E1yyNnIUrBcrRkJfsK4/bcz97hNTmkfyAWcx80IyljzglKCRBo1h2sLsMa26l9mwXHFqzgL2OnFzUoEczWH5azCKaBIyiVQQrfuuE6OXEoWcCpaVBolmMaFTMmZ9QyUJmfbSxc2ZfWGUkR1EypREe6H+nkhJqPUs9E1nSHCiV725+J/XTzC481Iu4wSZpMtFQSJsjOx5APaIK0ZRzAwhVHFzq00nRBGKJqaSCcFdfXmddK5q7k3NbV1X6tU8jiKcwTlUwYVbqMM9NKENFGJ4hld4sxLrxXq3PpatBSufOYU/sD5/ADzekRU=</latexit>

C(1)Qt

<latexit sha1_base64="pYWYeq3l0VMaQXKHF49NBQSvkx4=">AAAB83icbVBNTwIxEJ3iF+IX6tFLIzHBC9k1RjmScPEIiSAJrKRbutDQ7W7argnZ7N/w4kFjvPpnvPlvLLAHBV8yyct7M5mZ58eCa+M436iwsbm1vVPcLe3tHxwelY9PujpKFGUdGolI9XyimeCSdQw3gvVixUjoC/bgT5tz/+GJKc0jeW9mMfNCMpY84JQYKw2aw7Tdzh7Tav0yG5YrTs1ZAK8TNycVyNEalr8Go4gmIZOGCqJ133Vi46VEGU4Fy0qDRLOY0CkZs76lkoRMe+ni5gxfWGWEg0jZkgYv1N8TKQm1noW+7QyJmehVby7+5/UTE9S9lMs4MUzS5aIgEdhEeB4AHnHFqBEzSwhV3N6K6YQoQo2NqWRDcFdfXifdq5p7U3Pb15VGNY+jCGdwDlVw4RYacAct6ACFGJ7hFd5Qgl7QO/pYthZQPnMKf4A+fwARh5D5</latexit>

C(8)QQ

<latexit sha1_base64="Q3zA2Ni6rxiDEw8MChMvv26j2VQ=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRahXspGRD0WevHYgv2Adi3ZNNuGZrNLkhXKsn/DiwdFvPpnvPlvTNs9aOuDgcd7M8zM82PBtXHdb6ewsbm1vVPcLe3tHxwelY9POjpKFGVtGolI9XyimeCStQ03gvVixUjoC9b1p425331iSvNIPphZzLyQjCUPOCXGSoPGMG21sse0ii+zYbni1twF0DrBOalAjuaw/DUYRTQJmTRUEK372I2NlxJlOBUsKw0SzWJCp2TM+pZKEjLtpYubM3RhlREKImVLGrRQf0+kJNR6Fvq2MyRmole9ufif109McOelXMaJYZIuFwWJQCZC8wDQiCtGjZhZQqji9lZEJ0QRamxMJRsCXn15nXSuavimhlvXlXo1j6MIZ3AOVcBwC3W4hya0gUIMz/AKb07ivDjvzseyteDkM6fwB87nDwbdkPI=</latexit>

C(1)QQ

FIG. 5: Comparison of single parameter limits from loop corrections to EWPO involving 3rd

generation 4− fermion interactions with similar limits from LHC tt production[31].

In Fig. 6, we compare the current precision from the EWPO on the 3rd generation opera-

tors with that projected from a Tera-Z run at FCC-ee with an assumed integrated luminosity

of 150 ab−1 (3 × 1012 visible Z’s) and with a Giga-Z run at the ILC with an integrated lu-

minosity of 100 fb−1 (109 Z’s). This figure assumes that current theory uncertainties are

halved, assumes the FCC-ee precision of Table II in Ref. [35] and the ILC Giga-Z numbers

of Table 9 in Ref. [36]. Due to the polarization, for some observables the projected ILC

precision surpasses that of the FCC-ee, despite the smaller assumed luminosity.

IV. CONCLUSIONS

We have included flavor non-universal effects from 4−fermion operators with at least 2

quarks into the NLO electroweak and QCD corrections to the SMEFT predictions for the

precision electroweak observables. Our results are presented in a numerical form that can

be incorporated in the global fitting programs and suggest that the flavor assumptions on

the 4− fermion operators can have a significant effect. In particular we showed that the

bounds obtained from EWPO on the C(1)QQ and C

(1)Qt operators, that appear in the EWPO

only at NLO, are competitive with respect to those obtained from current LHC tt ob-

servables. Numerical results are posted at https://quark.phy.bnl.gov/Digital_Data_

9

-20

0

20

C/L

2 [TeV

-2]

LEP EWPOFCC-ee (Tera-Z)ILC (Giga-Z)

95% CL limits from NLO EWPO on 4-fermion operators

<latexit sha1_base64="dMw7pkyeaOPiEeE1fzn19nkf2QU=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkoioh4LvXhswX5AG8tms2mXbjZhdyOUkL/hxYMiXv0z3vw3btMctPXBwOO9GWbmeTFnStv2t1Xa2Nza3invVvb2Dw6PqscnPRUlktAuiXgkBx5WlDNBu5ppTgexpDj0OO17s9bC7z9RqVgkHvQ8pm6IJ4IFjGBtpFFrnHb87DGtO5fZuFqzG3YOtE6cgtSgQHtc/Rr5EUlCKjThWKmhY8faTbHUjHCaVUaJojEmMzyhQ0MFDqly0/zmDF0YxUdBJE0JjXL190SKQ6XmoWc6Q6ynatVbiP95w0QHd27KRJxoKshyUZBwpCO0CAD5TFKi+dwQTCQztyIyxRITbWKqmBCc1ZfXSe+q4dw0nM51rVkv4ijDGZxDHRy4hSbcQxu6QCCGZ3iFNyuxXqx362PZWrKKmVP4A+vzByQukQU=</latexit>

C(1)Qd

<latexit sha1_base64="znxeaz9A/ViN68AGXY+u4VnbKyA=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyURUY+FXjxWsLXQxrLZbNqlm03YnQgl5G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5fiK4Rsf5tkpr6xubW+Xtys7u3v5B9fCoq+NUUdahsYhVzyeaCS5ZBzkK1ksUI5Ev2IM/ac38hyemNI/lPU4T5kVkJHnIKUEjDVrDDIP8Mau75/mwWnMazhz2KnELUoMC7WH1axDENI2YRCqI1n3XSdDLiEJOBcsrg1SzhNAJGbG+oZJETHvZ/ObcPjNKYIexMiXRnqu/JzISaT2NfNMZERzrZW8m/uf1UwxvvIzLJEUm6WJRmAobY3sWgB1wxSiKqSGEKm5utemYKELRxFQxIbjLL6+S7kXDvWq4d5e1Zr2IowwncAp1cOEamnALbegAhQSe4RXerNR6sd6tj0VrySpmjuEPrM8fWlKRKA==</latexit>

C(1)td

<latexit sha1_base64="4GKPhwYgYtKTOitWeUOC+lOwnw4=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2FXRD0GcvGYgHlAsobZyWwyZHZ2mYcQlv0NLx4U8erPePNvnCR70MSChqKqm+6uIOFMadf9dgobm1vbO8Xd0t7+weFR+fiko2IjCW2TmMeyF2BFORO0rZnmtJdIiqOA024wbcz97hOVisXiQc8S6kd4LFjICNZWGjSGactkj2nVu8yG5YpbcxdA68TLSQVyNIflr8EoJiaiQhOOlep7bqL9FEvNCKdZaWAUTTCZ4jHtWypwRJWfLm7O0IVVRiiMpS2h0UL9PZHiSKlZFNjOCOuJWvXm4n9e3+jwzk+ZSIymgiwXhYYjHaN5AGjEJCWazyzBRDJ7KyITLDHRNqaSDcFbfXmddK5q3k3Na11X6tU8jiKcwTlUwYNbqMM9NKENBBJ4hld4c4zz4rw7H8vWgpPPnMIfOJ8/PmmRFg==</latexit>

C(1)Qu

<latexit sha1_base64="1rKyp2CdmPPgaDERMzioqhxLNqo=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkoioh4LvXisYGuhjWWz3bRLN5uwOxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzgkQKg6777RTW1jc2t4rbpZ3dvf2D8uFR28SpZrzFYhnrTkANl0LxFgqUvJNoTqNA8odg3Jj5D09cGxGre5wk3I/oUIlQMIpW6jX6GabTx6zqnU/75Ypbc+cgq8TLSQVyNPvlr94gZmnEFTJJjel6boJ+RjUKJvm01EsNTygb0yHvWqpoxI2fzW+ekjOrDEgYa1sKyVz9PZHRyJhJFNjOiOLILHsz8T+vm2J442dCJSlyxRaLwlQSjMksADIQmjOUE0so08LeStiIasrQxlSyIXjLL6+S9kXNu6p5d5eVejWPowgncApV8OAa6nALTWgBgwSe4RXenNR5cd6dj0VrwclnjuEPnM8fdI2ROQ==</latexit>

C(1)tu

<latexit sha1_base64="nG9H/hgp+benraBf9kShH6dbvZ0=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2FXRHMM5OIxgnlAsobZyWwyZHZ2mYcQlv0NLx4U8erPePNvnCR70MSChqKqm+6uIOFMadf9dgobm1vbO8Xd0t7+weFR+fiko2IjCW2TmMeyF2BFORO0rZnmtJdIiqOA024wbc797hOVisXiQc8S6kd4LFjICNZWGjSHqTbZY1qtX2bDcsWtuQugdeLlpAI5WsPy12AUExNRoQnHSvU9N9F+iqVmhNOsNDCKJphM8Zj2LRU4ospPFzdn6MIqIxTG0pbQaKH+nkhxpNQsCmxnhPVErXpz8T+vb3RY91MmEqOpIMtFoeFIx2geABoxSYnmM0swkczeisgES0y0jalkQ/BWX14nnauad1Pz7q8rjWoeRxHO4Byq4MEtNOAOWtAGAgk8wyu8OcZ5cd6dj2VrwclnTuEPnM8ffzeRQA==</latexit>

C(8)tu

<latexit sha1_base64="M4d1ZKW1yAFTEuXYyaVHj738hQw=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2FXRD0GcvEYwTwgWcPsZDYZMvtwplcIy/6GFw+KePVnvPk3TpI9aGJBQ1HVTXeXF0uh0ba/rcLa+sbmVnG7tLO7t39QPjxq6yhRjLdYJCPV9ajmUoS8hQIl78aK08CTvONNGjO/88SVFlF4j9OYuwEdhcIXjKKR+o1Bio/ZQ1p1zrNBuWLX7DnIKnFyUoEczUH5qz+MWBLwEJmkWvccO0Y3pQoFkzwr9RPNY8omdMR7hoY04NpN5zdn5MwoQ+JHylSIZK7+nkhpoPU08ExnQHGsl72Z+J/XS9C/cVMRxgnykC0W+YkkGJFZAGQoFGcop4ZQpoS5lbAxVZShialkQnCWX14l7Yuac1Vz7i4r9WoeRxFO4BSq4MA11OEWmtACBjE8wyu8WYn1Yr1bH4vWgpXPHMMfWJ8/bmGRNQ==</latexit>

C(1)tq

<latexit sha1_base64="TWxinwuME/kwuGprV7olHC5WMis=">AAAB9XicbVBNS8NAEJ34WetX1aOXxSJUkJKoqMdCLx5bsB/QpmWz3bRLN5u4u1FKyP/w4kERr/4Xb/4bt20O2vpg4PHeDDPzvIgzpW3721pZXVvf2Mxt5bd3dvf2CweHTRXGktAGCXko2x5WlDNBG5ppTtuRpDjwOG154+rUbz1SqVgo7vUkom6Ah4L5jGBtpF61n9Qf0l5Sujx3ztJ+oWiX7RnQMnEyUoQMtX7hqzsISRxQoQnHSnUcO9JugqVmhNM0340VjTAZ4yHtGCpwQJWbzK5O0alRBsgPpSmh0Uz9PZHgQKlJ4JnOAOuRWvSm4n9eJ9b+rZswEcWaCjJf5Mcc6RBNI0ADJinRfGIIJpKZWxEZYYmJNkHlTQjO4svLpHlRdq7LTv2qWCllceTgGE6gBA7cQAXuoAYNICDhGV7hzXqyXqx362PeumJlM0fwB9bnDxpIkYU=</latexit>

C(3,1)Qq

<latexit sha1_base64="AAQPjHpqp3D2AZxKcGJsAIFDbRk=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBahgpRERXss9OKxBfsBbVo22227dLOJuxulhPwPLx4U8ep/8ea/cdvmoK0PBh7vzTAzzws5U9q2v63M2vrG5lZ2O7ezu7d/kD88aqogkoQ2SMAD2fawopwJ2tBMc9oOJcW+x2nLm1RnfuuRSsUCca+nIXV9PBJsyAjWRupV+3H9IenFxauL8nnSzxfskj0HWiVOSgqQotbPf3UHAYl8KjThWKmOY4fajbHUjHCa5LqRoiEmEzyiHUMF9qly4/nVCTozygANA2lKaDRXf0/E2Fdq6num08d6rJa9mfif14n0sOzGTISRpoIsFg0jjnSAZhGgAZOUaD41BBPJzK2IjLHERJugciYEZ/nlVdK8LDk3Jad+XagU0ziycAKnUAQHbqECd1CDBhCQ8Ayv8GY9WS/Wu/WxaM1Y6cwx/IH1+QMk8pGM</latexit>

C(3,8)Qq

<latexit sha1_base64="FOoMwfj19yFA6vakqA5CawiTdz8=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69BItQQUpWRD0WevHYgv2ANi2b7aZdutnE3Y1SQv6HFw+KePW/ePPfuG1z0NYHA4/3ZpiZ50WcKe0431ZubX1jcyu/XdjZ3ds/KB4etVQYS0KbJOSh7HhYUc4EbWqmOe1EkuLA47TtTWozv/1IpWKhuNfTiLoBHgnmM4K1kfq1QdJ4SPtJGV2g83RQLDkVZw57laCMlCBDfVD86g1DEgdUaMKxUl3kRNpNsNSMcJoWerGiESYTPKJdQwUOqHKT+dWpfWaUoe2H0pTQ9lz9PZHgQKlp4JnOAOuxWvZm4n9eN9b+rZswEcWaCrJY5Mfc1qE9i8AeMkmJ5lNDMJHM3GqTMZaYaBNUwYSAll9eJa3LCrquoMZVqVrO4sjDCZxCGRDcQBXuoA5NICDhGV7hzXqyXqx362PRmrOymWP4A+vzBxc4kYM=</latexit>

C(1,1)Qq

<latexit sha1_base64="HFqmPYOgN9neiGupk4USY7Et4qA=">AAAB9XicbVBNS8NAEJ34WetX1aOXxSJUkJKIaI+FXjy2YD+gTctmu2mXbjZxd6OUkP/hxYMiXv0v3vw3btsctPXBwOO9GWbmeRFnStv2t7W2vrG5tZ3bye/u7R8cFo6OWyqMJaFNEvJQdjysKGeCNjXTnHYiSXHgcdr2JrWZ336kUrFQ3OtpRN0AjwTzGcHaSP3aIGk8pP2k5FxWLtJBoWiX7TnQKnEyUoQM9UHhqzcMSRxQoQnHSnUdO9JugqVmhNM034sVjTCZ4BHtGipwQJWbzK9O0blRhsgPpSmh0Vz9PZHgQKlp4JnOAOuxWvZm4n9eN9Z+xU2YiGJNBVks8mOOdIhmEaAhk5RoPjUEE8nMrYiMscREm6DyJgRn+eVV0roqOzdlp3FdrJayOHJwCmdQAgduoQp3UIcmEJDwDK/wZj1ZL9a79bFoXbOymRP4A+vzByHikYo=</latexit>

C(1,8)Qq

<latexit sha1_base64="Z5zPwFXgu1VQDE7xsBnzCNCH2oE=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyURUY+FXjy2YD+gjWWz3bRLN5uwOxFKyN/w4kERr/4Zb/4bt20O2vpg4PHeDDPz/FhwjY7zbRU2Nre2d4q7pb39g8Oj8vFJR0eJoqxNIxGpnk80E1yyNnIUrBcrRkJfsK4/bcz97hNTmkfyAWcx80IyljzglKCRBo1h2sLsMa26l9mwXHFqzgL2OnFzUoEczWH5azCKaBIyiVQQrfuuE6OXEoWcCpaVBolmMaFTMmZ9QyUJmfbSxc2ZfWGUkR1EypREe6H+nkhJqPUs9E1nSHCiV725+J/XTzC481Iu4wSZpMtFQSJsjOx5APaIK0ZRzAwhVHFzq00nRBGKJqaSCcFdfXmddK5q7k3NbV1X6tU8jiKcwTlUwYVbqMM9NKENFGJ4hld4sxLrxXq3PpatBSufOYU/sD5/ADzekRU=</latexit>

C(1)Qt

<latexit sha1_base64="pYWYeq3l0VMaQXKHF49NBQSvkx4=">AAAB83icbVBNTwIxEJ3iF+IX6tFLIzHBC9k1RjmScPEIiSAJrKRbutDQ7W7argnZ7N/w4kFjvPpnvPlvLLAHBV8yyct7M5mZ58eCa+M436iwsbm1vVPcLe3tHxwelY9PujpKFGUdGolI9XyimeCSdQw3gvVixUjoC/bgT5tz/+GJKc0jeW9mMfNCMpY84JQYKw2aw7Tdzh7Tav0yG5YrTs1ZAK8TNycVyNEalr8Go4gmIZOGCqJ133Vi46VEGU4Fy0qDRLOY0CkZs76lkoRMe+ni5gxfWGWEg0jZkgYv1N8TKQm1noW+7QyJmehVby7+5/UTE9S9lMs4MUzS5aIgEdhEeB4AHnHFqBEzSwhV3N6K6YQoQo2NqWRDcFdfXifdq5p7U3Pb15VGNY+jCGdwDlVw4RYacAct6ACFGJ7hFd5Qgl7QO/pYthZQPnMKf4A+fwARh5D5</latexit>

C(8)QQ

<latexit sha1_base64="Q3zA2Ni6rxiDEw8MChMvv26j2VQ=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRahXspGRD0WevHYgv2Adi3ZNNuGZrNLkhXKsn/DiwdFvPpnvPlvTNs9aOuDgcd7M8zM82PBtXHdb6ewsbm1vVPcLe3tHxwelY9POjpKFGVtGolI9XyimeCStQ03gvVixUjoC9b1p425331iSvNIPphZzLyQjCUPOCXGSoPGMG21sse0ii+zYbni1twF0DrBOalAjuaw/DUYRTQJmTRUEK372I2NlxJlOBUsKw0SzWJCp2TM+pZKEjLtpYubM3RhlREKImVLGrRQf0+kJNR6Fvq2MyRmole9ufif109McOelXMaJYZIuFwWJQCZC8wDQiCtGjZhZQqji9lZEJ0QRamxMJRsCXn15nXSuavimhlvXlXo1j6MIZ3AOVcBwC3W4hya0gUIMz/AKb07ivDjvzseyteDkM6fwB87nDwbdkPI=</latexit>

C(1)QQ

FIG. 6: Comparison of single parameter limits from loop corrections to EWPO involving 3rd

generation 4− fermion interactions with projected Z pole limits from a Tera-Z program at the

FCC-ee[35] and with a Giga-Z ILC run[36] .

Archive/dawson/ewpo_22.

Acknowledgements

We thank Susanne Westhoff, Danny van Dyk, and Sebastian Bruggisse for pointing out

that including non-trivial flavor dependencies in the SMEFT NLO predictions for the EWPO

could be important for the global fits and also for valuable comments on the manuscript.

S.D. is supported by the U.S. Department of Energy under Grant Contract de-sc0012704.

The work of PPG has received financial support from Xunta de Galicia (Centro singular

de investigacion de Galicia accreditation 2019-2022), by European Union ERDF, and by

“Marıa de Maeztu” Units of Excellence program MDM-2016-0692 and the Spanish Research

State Agency.

[1] S. Dawson and P. P. Giardino, “Electroweak and QCD corrections to Z and W pole

observables in the standard model EFT,” Phys. Rev. D 101 no. 1, (2020) 013001,

arXiv:1909.02000 [hep-ph].

10

[2] T. Corbett, O. J. P. Eboli, and M. C. Gonzalez-Garcia, “Unitarity Constraints on

Dimension-six Operators II: Including Fermionic Operators,” Phys. Rev. D96 no. 3, (2017)

035006, arXiv:1705.09294 [hep-ph].

[3] L. Berthier and M. Trott, “Consistent constraints on the Standard Model Effective Field

Theory,” JHEP 02 (2016) 069, arXiv:1508.05060 [hep-ph].

[4] T. Corbett, A. Helset, A. Martin, and M. Trott, “EWPD in the SMEFT to dimension

eight,” JHEP 06 (2021) 076, arXiv:2102.02819 [hep-ph].

[5] J. de Blas et al., “Higgs Boson Studies at Future Particle Colliders,” JHEP 01 (2020) 139,

arXiv:1905.03764 [hep-ph].

[6] S. Dawson, S. Homiller, and S. D. Lane, “Putting standard model EFT fits to work,” Phys.

Rev. D 102 no. 5, (2020) 055012, arXiv:2007.01296 [hep-ph].

[7] C. Degrande, G. Durieux, F. Maltoni, K. Mimasu, E. Vryonidou, and C. Zhang, “Automated

one-loop computations in the standard model effective field theory,” Phys. Rev. D 103 no. 9,

(2021) 096024, arXiv:2008.11743 [hep-ph].

[8] J. M. Cullen and B. D. Pecjak, “Higgs decay to fermion pairs at NLO in SMEFT,” JHEP 11

(2020) 079, arXiv:2007.15238 [hep-ph].

[9] J. M. Cullen, B. D. Pecjak, and D. J. Scott, “NLO corrections to h→ bb decay in SMEFT,”

arXiv:1904.06358 [hep-ph].

[10] R. Gauld, B. D. Pecjak, and D. J. Scott, “QCD radiative corrections for h→ bb in the

Standard Model Dimension-6 EFT,” Phys. Rev. D94 no. 7, (2016) 074045,

arXiv:1607.06354 [hep-ph].

[11] C. Hartmann and M. Trott, “Higgs Decay to Two Photons at One Loop in the Standard

Model Effective Field Theory,” Phys. Rev. Lett. 115 no. 19, (2015) 191801,

arXiv:1507.03568 [hep-ph].

[12] C. Hartmann and M. Trott, “On one-loop corrections in the standard model effective field

theory; the Γ(h→ γ γ) case,” JHEP 07 (2015) 151, arXiv:1505.02646 [hep-ph].

[13] S. Dawson and P. P. Giardino, “Electroweak corrections to Higgs boson decays to γγ and

W+W− in standard model EFT,” Phys. Rev. D98 no. 9, (2018) 095005, arXiv:1807.11504

[hep-ph].

[14] A. Dedes, M. Paraskevas, J. Rosiek, K. Suxho, and L. Trifyllis, “The decay h→ γγ in the

Standard-Model Effective Field Theory,” JHEP 08 (2018) 103, arXiv:1805.00302

11

[hep-ph].

[15] S. Dawson and P. P. Giardino, “Higgs decays to ZZ and Zγ in the standard model effective

field theory: An NLO analysis,” Phys. Rev. D97 no. 9, (2018) 093003, arXiv:1801.01136

[hep-ph].

[16] A. Dedes, K. Suxho, and L. Trifyllis, “The decay h→ Zγ in the Standard-Model Effective

Field Theory,” JHEP 06 (2019) 115, arXiv:1903.12046 [hep-ph].

[17] C. Hartmann, W. Shepherd, and M. Trott, “The Z decay width in the SMEFT: yt and λ

corrections at one loop,” JHEP 03 (2017) 060, arXiv:1611.09879 [hep-ph].

[18] R. Boughezal, C.-Y. Chen, F. Petriello, and D. Wiegand, “Top quark decay at

next-to-leading order in the Standard Model Effective Field Theory,” arXiv:1907.00997

[hep-ph].

[19] S. Dawson, P. P. Giardino, and A. Ismail, “Standard model EFT and the Drell-Yan process

at high energy,” Phys. Rev. D 99 no. 3, (2019) 035044, arXiv:1811.12260 [hep-ph].

[20] S. Dawson and P. P. Giardino, “New physics through Drell-Yan standard model EFT

measurements at NLO,” Phys. Rev. D 104 no. 7, (2021) 073004.

[21] S. Bruggisser, R. Schafer, D. van Dyk, and S. Westhoff, “The Flavor of UV Physics,” JHEP

05 (2021) 257, arXiv:2101.07273 [hep-ph].

[22] L. Alasfar, A. Azatov, J. de Blas, A. Paul, and M. Valli, “B anomalies under the lens of

electroweak precision,” JHEP 12 (2020) 016, arXiv:2007.04400 [hep-ph].

[23] E. d. S. Almeida, A. Alves, O. J. P. Eboli, and M. C. Gonzalez-Garcia, “Electroweak legacy

of the LHC run II,” Phys. Rev. D 105 no. 1, (2022) 013006, arXiv:2108.04828 [hep-ph].

[24] S. Bißmann, C. Grunwald, G. Hiller, and K. Kroninger, “Top and Beauty synergies in

SMEFT-fits at present and future colliders,” JHEP 06 (2021) 010, arXiv:2012.10456

[hep-ph].

[25] SMEFiT Collaboration, J. J. Ethier, G. Magni, F. Maltoni, L. Mantani, E. R. Nocera,

J. Rojo, E. Slade, E. Vryonidou, and C. Zhang, “Combined SMEFT interpretation of Higgs,

diboson, and top quark data from the LHC,” JHEP 11 (2021) 089, arXiv:2105.00006

[hep-ph].

[26] J. Ellis, M. Madigan, K. Mimasu, V. Sanz, and T. You, “Top, Higgs, Diboson and

Electroweak Fit to the Standard Model Effective Field Theory,” JHEP 04 (2021) 279,

arXiv:2012.02779 [hep-ph].

12

[27] I. Brivio and M. Trott, “The Standard Model as an Effective Field Theory,” Phys. Rept.

793 (2019) 1–98, arXiv:1706.08945 [hep-ph].

[28] W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions and

Flavor Conservation,” Nucl. Phys. B 268 (1986) 621–653.

[29] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, “Dimension-Six Terms in the

Standard Model Lagrangian,” JHEP 10 (2010) 085, arXiv:1008.4884 [hep-ph].

[30] J. J. Ethier, G. Magni, F. Maltoni, L. Mantani, E. R. Nocera, J. Rojo, E. Slade,

E. Vryonidou, and C. Zhang, “Combined smeft interpretation of higgs, diboson, and top

quark data from the lhc,” 2021.

[31] C. Zhang, “Constraining qqtt operators from four-top production: a case for enhanced EFT

sensitivity,” Chin. Phys. C 42 no. 2, (2018) 023104, arXiv:1708.05928 [hep-ph].

[32] I. Brivio, S. Bruggisser, F. Maltoni, R. Moutafis, T. Plehn, E. Vryonidou, S. Westhoff, and

C. Zhang, “O new physics, where art thou? A global search in the top sector,” JHEP 02

(2020) 131, arXiv:1910.03606 [hep-ph].

[33] A. Buckley, C. Englert, J. Ferrando, D. J. Miller, L. Moore, M. Russell, and C. D. White,

“Global fit of top quark effective theory to data,” Phys. Rev. D 92 no. 9, (2015) 091501,

arXiv:1506.08845 [hep-ph].

[34] D. Barducci et al., “Interpreting top-quark LHC measurements in the standard-model

effective field theory,” arXiv:1802.07237 [hep-ph].

[35] A. Blondel et al., “Standard model theory for the FCC-ee Tera-Z stage,” in Mini Workshop

on Precision EW and QCD Calculations for the FCC Studies : Methods and Techniques,

vol. 3/2019 of CERN Yellow Reports: Monographs. CERN, Geneva, 9, 2018.

arXiv:1809.01830 [hep-ph].

[36] K. Fujii, C. Grojean, M. E. Peskin, T. Barklow, Y. Gao, S. Kanemura, H. Kim, J. List,

M. Nojiri, M. Perelstein, R. Poeschl, J. Reuter, F. Simon, T. Tanabe, J. D. Wells, J. Yu,

J. Tian, T. Suehara, M. Vos, G. Wilson, J. Brau, and H. Murayama, “Tests of the standard

model at the international linear collider,” 2019.

13

No flavor dep. Only 3rd gen. At least 3rd gen. Only 1st, 2nd gen.

C(1)qq

Λ2 [-0.93,1.48] [-0.80,1.34] [-0.93,1.54] [-11.45,10.81]

C(3)qq

Λ2 [-0.32,0.29] [-9.01,15.02] [-0.49,0.42] [-0.94,.0.88]

C(1)qu

Λ2 [-2.17,1.33] [-2.24,1.34] [-2.20,1.35] [-46.61,42.53]

C(1)qd

Λ2 [-9.76,4.98] [-21.00.,4.87] [-8.98,4.79] [-85.71,90.22]

CuuΛ2 [-1.14,0.99] - [-1.20,1.04] [-22.56,19.48]

CddΛ2 [-50.60,26.29] [-364.80,94.77] [-89.93,31.77] [-87.75,82.43]

C(1)ud

Λ2 [-3.01,5.62] [-4.06,15.62] [-3.17,6.21] [-45.74,50.83]

C(1)lq

Λ2 [-0.25,0.66] [-0.24,0.63] [-0.24,0.63] [-13.25,5.44]

C(3)lq

Λ2 [-0.32,0.57] [-0.29,0.68] [-0.29,0.68] [-1.92,1.02]

CluΛ2 [-0.49,0.19] [-0.53,0.21] [-0.53,0.21] [-6.62,2.74]

CldΛ2 [-3.76,8.71] [-11.99,25.13] [-11.99,25.13] [-5.45,13.28]

Cqe

Λ2 [-0.75,0.48] [-0.72,0.45] [-0.72,0.45] [-10.50,17.30]

CedΛ2 [-11.80,6.71] [-36.23,18.28] [-36.23,18.28] [-17.41,10.52]

CeuΛ2 [-0.36,0.58] [-0.39,0.62] [-0.39,0.62] [-5.23,8.72]

TABLE II: 95% CL single parameter limits in TeV −2 from EWPO in the Warsaw basis. The

second column corresponds to the NLO results in our previous paper with no flavor dependence

in the quark and lepton sectors, the 3rd column has the only non-zero contributions from the 3rd

generation fermions, the 4th column sets coefficients which only involve the 1st and 2nd generation

quarks to 0, and the 5th column has equal contributions for each coefficient from operators involving

the 1st and 2nd generations, with no contributions from the 3rd generation.

14

Operator EWPO tt(1/Λ2) tt(1/Λ4)

C(1)QQ

Λ2 ≡ 2C

(1)qq,[3333]

Λ2 − 23

C(3)qq,[3333]

Λ2 [-1.61,2.68] [-6.132, 23.281] [-2.229,2.019]

C(8)QQ

Λ2 ≡ 8C

(3)qq,[3333]

Λ2 [-15.23,25.41] [-26.471,57.778] [-6.812,5.834]

C(1)Qt

Λ2 ≡C

(1)qu,[3333]

Λ2 [-2.24,1.35] [-195,159] [-1.830,1.862]

C(1,8)Qq

Λ2 ≡C

(1)qq,[i33i]

Λ2 + 3C

(3)qq,[i33i]

Λ2 [-18.67,20.19] [-0.273,0.509] [-0.373,0.309]

C(1,1)Qq

Λ2 ≡C

(1)qq,[ii33]

Λ2 + 16

C(1)qq,[i33i]

Λ2 + 12

C(3)qq,[i33i]

Λ2 [-3.47,3.36] [-3.603,0.307] [-0.303,0.225]

C(3,8)Qq

Λ2 ≡C

(1)qq,[i33i]

Λ2 −C

(3)qq,[i33i]

Λ2 [-3.03, 3.04] [-1.813,0.625] [-0.470,0.439]

C(3,1)Qq

Λ2 ≡C

(3)qq,[ii33]

Λ2 + 16

(C

(1)qq,[i33i]

Λ2 −C

(3)qq,[i33i]

Λ2

)[-0.32, 0.28] [-0.099,0.155] [-0.088,0.166]

C(1)tq

Λ2 ≡C

(1)qu,[ii33]

Λ2 [-5.74,5.52] [-0.784,2.771] [-0.205,0.271]

C(8)tu

Λ2 ≡ 2Cuu,[i33i]

Λ2 [-14.28,12.33] [-0.774,0.607] [-0.911,0.347]

C(1)tu

Λ2 ≡Cuu,[ii33]

Λ2 + 13

Cuu,[i33i]

Λ2 [-1.93,1.67] [-6.046,0.424] [-0.380,0.293]

C(1)Qu

Λ2 ≡C

(1)qu,[33ii]

Λ2 [-4.40,4.67] [-0.938,2.462] [-0.281,0.371]

C(1)td

Λ2 ≡C

(1)ud,[33jj]

Λ2 [-3.38,6.50] [-9.504,-0.086] [-0.449,0.371]

C(1)Qd

Λ2 ≡C

(1)qd,[33jj]

Λ2 [-8.70,4.59] [-0.889,6.459] [-0.332,0.436]

TABLE III: 95% CL single parameter limits in TeV −2 from NLO contributions of 3rd generation

4-fermion operators to EWPO (1st column) to LHC tt results tt results using the O( 1Λ2 ) and O( 1

Λ4 )

expansions[30]. The generation index i = 1, 2 and j = 1, 2, 3 and the scale Λ = 1 TeV .

15