# Computer package DIZET v. 6.45

A. Arbuzov  Dubna, 141980 Russia J. Gluza  Chorzów, 41-500 Poland  L.
Kalinovskaya  Dubna, 141980 Russia S. Riemann  Königs Wusterhausen, D-15711
Germany  T. Riemann  Königs Wusterhausen, D-15711 Germany  V. Yermolchyk
Dubna, 141980 Russia Minsk, 220006 Belarus

###### Abstract

The new version of the DIZET electroweak library is described. Changes and
additional code features concerning the previous version are explained. The
software allows one to make state-of-the-art theoretical predictions for
pseudo-observable quantities, including higher-order radiative corrections.
The current version of the DIZET library v. 6.45 incorporates advanced recent
results of theoretical calculations. Numerical comparisons with the results of
the previous version are performed. Estimates of theoretical uncertainties are
discussed.

##  1 Introduction

The computer package DIZET was created as electroweak and QCD library of the
ZFITTER program [2, 4] which was one of the main tools for the high-precision
verification of the Standard Model at LEP [6]. DIZET can also be linked as a
library by other projects, e.g., it is used by the HECTOR program [8] and by
the KKMC Monte Carlo event generator [10].

DIZET can be used for fitting EWPOs, for instance $\sin^{2}\vartheta_{eff}$ at
LHC as discussed at the LHC EW precision workshop [12]. In each new version of
DIZET, the compatibility with all previous versions has been preserved. Thus,
the numerics of a previous version can be fully reproduced, except for changes
caused by the correction of bugs. The latter is documented in the header of
the code. We remind, that between DIZET versions 6.21 [2] and 6.42 [4], there
are changes affecting the $W$ boson width and the running of the
electromagnetic coupling $\alpha_{QED}$. In addition, the treatment of so-
called box-like diagram contributions (controlled by the IBOXF flag of the
ROKANC subroutine in ZFITTER) and of the $b$ quark production channel (IBFLA
flag in the same subroutine) have been changed. Links to publications and the
public versions of DIZET can be found at the ZFITTER project webpage
http://sanc.jinr.ru/users/zfitter/.

The last documented version of DIZET is 6.42 [4]. In the present paper, we
describe the actual DIZET version 6.45 [14]. We present the transition from
DIZET v. 6.42 to DIZET v. 6.45 and show the numerical impact of the newly
introduced modifications, controlled by the corresponding options and flags on
pseudo-observables (EWPOs). The contributions added in DIZET v. 6.45 are
connected with the completion of the 2-loop EW radiative corrections given in
[16, 18] and which complement earlier works on radiative corrections, namely:
the complete fermionic two-loop corrections to the $W$ boson mass [20]; the
leading ${\cal O}(\alpha\alpha_{s})$ [22] and next-to-leading ${\cal
O}(\alpha\alpha_{s}^{2})$ [24, 26, 28] QCD corrections, as well as leading
three-loop corrections in an expansion in $m_{t}^{2}$ of order ${\cal
O}(\alpha^{3})$ and ${\cal O}(\alpha^{2}\alpha_{s})$ [30].

These modifications are relevant for future precision HL-LHC studies and the
LHC electroweak Working Group activities. They are also needed as a first step
towards high precision predictions of the Standard Model electroweak effects
at future high energy colliders. In the context of the future circular
electron-positron collider (FCC-ee) [32], the anticipated experimental
accuracy on EWPOs has to be matched with theory predictions of at least the
same level of accuracy to achieve maximum usage of experimental data. For the
present situation concerning EWPOs determination and their future estimate,
see Tab. 1 and references [34, 36] for more details. In Tab. 1, we put
experimental predictions for EWPOs at FCC-ee as the most stringent among
future experimental setups, particularly in the Z-resonance region. Other
widely considered future $e^{+}e^{-}$ collider projects are CEPC [38], ILC
[40, 42], and CLIC [44, 46]. Through their high integrated luminosities of
several ab-1 (practically for all relevant Z-resonance, $HZ$, $WW$, and
$t\bar{t}$ modes [32, 38]) these machines will be sensitive to very small
deviations between the measured value and the SM expectation for a given
observable. To account correctly for such slight deviations, dedicated
programs like here discussed DIZET will be highly needed. Table 1 shows the
comparison between the estimated FCC-ee experimental precision, the current
theoretical uncertainty, and the so-called projected one for representative
EWPOs, see Chapter B in [34] and [36]. By the projected theoretical
uncertainty we mean an estimate of the future theoretical uncertainty when the
leading 3-loop
$\mathcal{O}(\alpha^{3},\alpha^{2}\alpha_{s},\alpha\alpha_{s}^{2})$
corrections will become available.

Quantity | FCC-ee | Current theory | Projected theory
---|---|---|---
|  | uncertainty | uncertainty
$m_{\mathrm{W}}$ (MeV) | $0.5-1$ | 4 | 1
$\sin^{2}\vartheta^{\ell}_{\rm{eff}}$ ($10^{-5}$) | 0.6 | 4.5 | 1.5
$\Gamma_{\mathrm{Z}}$ (MeV) | 0.1 | 0.4 | 0.15
$R_{b}$ ($10^{-5}$) | 6 | 11 | 5
$R_{\ell}$ ($10^{-3}$) | 1 | 6 | 1.5
Table 1: Estimated precision for the direct determination of representative
EWPOs at FCC-ee (column 2), current theory uncertainties for the SM prediction
of these quantities (column 3), and the projected theoretical uncertainty
(column 4).

Indeed, as DIZET includes Standard Model higher order radiative corrections,
it can be used for comparisons with experimental results in search for models
which go beyond the Standard Model. We discuss here the impact of newly
implemented corrections in DIZET on EWPOs and form-factors at the $e^{+}e^{-}$
resonance.

##  2 Release DIZET from v. 6.42 to v. 6.45

The Fortran code DIZET is a library for the calculation of electroweak
radiative corrections and it is part of the ZFITTER distribution package. It
can also be used in a stand-alone mode.

On default, DIZET performs the following calculations:

  * •

by call of subroutine ROKANC: four weak neutral-current (NC) form factors,
running electromagnetic and strong couplings needed for the calculation of
effective NC Born cross sections for the production of massless fermions
(however, the mass of the top quark appearing in the virtual state of loop
diagrams for the process $e^{+}e^{-}\to f\bar{f}$ is not ignored);

  * •

by call of subroutine RHOCC: the corresponding form factors and running strong
coupling for the calculation of effective CC Born cross sections;

  * •

by call of subroutine ZU_APV: $Q_{W}(Z,A)$ – the weak charge used for the
description of parity violation in heavy atoms.

If needed, the form factors of cross sections may be made to contain the
contributions from $WW$ and $ZZ$ box diagrams thus ensuring the correct
kinematic behaviour over a larger energy range compared to the Z pole.

Between the DIZET versions 6.42 and 6.45 there are changes affecting the
running of the QED coupling $\alpha(s)$ and the QCD corrections of order
$\alpha\alpha_{S}$ to the $Z$ boson partial widths. Also, starting from
v.6.44, DIZET uses the complete $\alpha_{s}^{4}$ QCD corrections to hadronic
$Z$-decays [48] by default. In order to reproduce the old behavior, the
IBAIKOV flag must be set to 2014 in the code of DIZET v. 6.45. The largest
contribution of the electroweak (EW) corrections comes from the $s$ channel
QED running of $\alpha(s)$, and the main load in it is due to the hadronic
component $\Delta\alpha^{(5)}_{had}(M_{\scriptscriptstyle{Z}})$ [50].

###  2.1 New options in DIZET v. 6.45

In this section, we give the descriptions of flags and added options
implemented in DIZET v. 6.45.

$\bullet$ flag IAMT4:

IAMT4: two-loop $\alpha^{2}$ bosonic and/or fermionic radiative corrections:

IVALUE = I

AVALUE =6

— fermionic two-loop corrections to $\sin^{2}\vartheta_{\rm eff}^{l}$ [52];

AVALUE =7

— the complete two-loop corrections to $\sin^{2}\vartheta_{\rm eff}^{b}$ and
$\sin^{2}\vartheta_{\rm eff}^{l}$ according to Refs. [16, 54].

AVALUE =8

— the complete electroweak two-loop radiative corrections to all the relevant
electroweak precision pseudo-observables related to the Z-boson, according to
Ref. [18].

The complete set of EWPOs related to the $Z$-boson for IAMT4 = 8 includes: the
leptonic and bottom-quark effective weak mixing angles $\sin^{2}\vartheta_{\rm
eff}^{\ell}$, $\sin^{2}\vartheta_{\rm eff}^{b}$, the $Z$-boson partial decay
widths $\Gamma_{f}$, where $f$ indicates any charged lepton, neutrino and
quark flavor (except for the top quark), the total $Z$ decay width
$\Gamma_{Z}$, the branching ratios $R_{\ell}$, $R_{c}$, $R_{b}$, and the
hadronic cross section $\sigma_{\rm had}^{0}$.

$\bullet$ flag IHVP:

IHVP — choice of hadronic vacuum polarization
$\Delta\alpha^{(5)}_{had}(M_{\scriptscriptstyle{Z}})$ using public versions of
the AlphaQED code by F. Jegerlehner:

IVALUE = I

AVALUE =1

— realization of the fit given in [56],

AVALUE =4

— realization of the fit by [58],

AVALUE =5

— realization of the fit by [60].

Details on hadronic vacuum polarization effects can be found in [60].

IHVP | 1 | 4 | 5
---|---|---|---
version | FJ-1995 | FJ-2016 | FJ-2017
$\Delta\alpha^{(5)}_{had}(M_{\scriptscriptstyle{Z}})$ | 2.8039e-2 | 2.7586e-2 | 2.7576e-2
Table 2: Results of the fit for hadronic vacuum polarization
$\Delta\alpha^{(5)}_{had}(M_{\scriptscriptstyle{Z}})$ for different versions
(1995-[56], 2016-[58], 2017-[60])

Note that the AlphaQED (2017) code provides an estimation of statistical and
systematic errors. To estimate the resulting uncertainties of a DIZET output
one has to run the code in a cycle with variation of the input parameters such
as the top quark and Higgs boson masses within their error bars, see, e.g.,
Ref. [62]. In addition one has to estimate the missing contributions of not
yet computed higher order corrections.

##  3 Numerical results

All numbers presented below are obtained with the following set of Input
Parameters (IPS) and their variations within experimental errors taken from
PDG Summary Tables, [64]: $\alpha^{-1}(0)=137.035999084$,
$\alpha_{s}(M_{Z})=0.1179$, $M_{Z}=91.1876\;\mathrm{GeV}$,
$M_{H}=125.25\;\mathrm{GeV}$, $m_{t}=172.76\;\mathrm{GeV}$. The masses of the
five light quarks are chosen in the usual way to reproduce the hadronic
contribution to the photon vacuum polarization (relevant only for $IHVP=2$)
$\Delta\alpha^{(5)}_{had}(M_{\scriptscriptstyle{Z}})$. Here numerical
calculations were carried out at the fixed value IBAIKOV=2014.

The numerical results presented here are slightly different from those of our
report [12] due to the change of input parameters. The present values of
pseudo-observables (EW boson widths and the weak mixing angle) are [64]:
$\Gamma_{Z}=2495.200\pm 2.300$ MeV, $G_{Z}(\mu\mu)=83.99\pm 0.16$ MeV,
$\Gamma_{W}=2085\pm 42$ MeV, $G_{W}(l\nu)=226.4\pm 1.9$ MeV,
$\sin^{2}\vartheta_{eff}\times 10^{6}=231480\pm 160$.

###  3.1 Parametric uncertainties

DIZET can calculate pseudo-observables and EW form-factors in a wide range of
input parameters: $M_{\scriptscriptstyle{H}}$, $M_{\scriptscriptstyle{Z}}$,
$m_{t}$, $\alpha_{s}$. Tables 3 \- 6 present the dependence of pseudo-
observables on the experimental uncertainty of input parameters:
($m_{t}=172.76(0.30)\,\mathrm{GeV}$,
$M_{\scriptscriptstyle{H}}=125.25(0.17)\,\mathrm{GeV}$,
$M_{\scriptscriptstyle{Z}}=91.1876(0.0021)\,\mathrm{GeV}$,
$\alpha_{s}=0.1179(0.0009)$). The first type of theoretical uncertainties are
due to variation of input parameters within experimental errors. We consider
first the parametric uncertainties due to variation of the masses $m_{t}$,
$M_{\scriptscriptstyle{H}}$ and $M_{\scriptscriptstyle{Z}}$ and in addition
the $\alpha_{s}$-dependence.

$m_{t},~\,\mathrm{GeV}$ | $172.76-0.30$ | 172.76 | $172.76+0.30$ | Diff.
---|---|---|---|---
$G_{Z}(\mu\mu),~\,\mathrm{MeV}$ | 83.982 | 83.985 | 83.987 | 0.005
$\Gamma_{Z},~\,\mathrm{MeV}$ | 2494.746 | 2494.814 | 2494.883 | 0.137
$G_{W}(l\nu),~\,\mathrm{MeV}$ | 678.935 | 678.981 | 679.027 | 0.092
$\Gamma_{W},~\,\mathrm{MeV}$ | 2089.825 | 2089.967 | 2090.109 | 0.284
$\sin^{2}\vartheta^{l}_{eff}\times 10^{6}$ | 231508 | 231500 | 231491 | 17
Table 3: The effect of the parametric uncertainty in $m_{t}$ on the magnitudes
of pseudo-observables.

As one can see, the parametric uncertainties for the listed pseudo-observables
are less than the current experimental errors [64].

$M_{\scriptscriptstyle{H}},~\,\mathrm{GeV}$ | $125.25-0.17$ | 125.25 | $125.25+0.17$ | Diff.
---|---|---|---|---
$G_{\scriptscriptstyle{Z}}(\mu\mu),~\,\mathrm{MeV}$ | 83.985 | 83.985 | 83.985 | 0
$\Gamma_{\scriptscriptstyle{Z}},~\,\mathrm{MeV}$ | 2494.818 | 2494.814 | 2494.811 | 0.007
$G_{\scriptscriptstyle{W}}(l\nu),~\,\mathrm{MeV}$ | 678.983 | 678.981 | 678.979 | 0.004
$\Gamma_{\scriptscriptstyle{W}},~\,\mathrm{MeV}$ | 2089.973 | 2089.967 | 2089.961 | 0.012
$\sin^{2}\vartheta^{l}_{eff}\times 10^{6}$ | 231499 | 231500 | 231500 | 1
Table 4:  The effect of the parametric uncertainty in
$M_{\scriptscriptstyle{H}}$ on the magnitudes of pseudo-observables.

Tables 3 and 4 show that the effect of experimental uncertainty of $m_{t}$ and
$M_{\scriptscriptstyle{H}}$ changes the partial widths by an interval not
exceeding their experimental errors.

$M_{\scriptscriptstyle{Z}},~\,\mathrm{GeV}$ | $91.1876-0.0021$ | 91.1876 | $91.1876+0.0021$ | Diff.
---|---|---|---|---
$G_{\scriptscriptstyle{Z}}(\mu\mu),~\,\mathrm{MeV}$ | 83.978 | 83.985 | 83.991 | 0.013
$\Gamma_{\scriptscriptstyle{Z}},~\,\mathrm{MeV}$ | 2494.602 | 2494.814 | 2495.027 | 0.425
$G_{\scriptscriptstyle{W}}(l\nu),~\,\mathrm{MeV}$ | 678.914 | 678.981 | 679.048 | 0.287
$\Gamma_{\scriptscriptstyle{W}},~\,\mathrm{MeV}$ | 2089.761 | 2089.967 | 2090.173 | 0.412
$\sin^{2}\vartheta^{l}_{eff}\times 10^{6}$ | 231515 | 231500 | 231485 | 30
Table 5: The effect of the parametric uncertainty in $M_{\scriptscriptstyle{Z}}$ on the magnitudes of pseudo-observables. $\alpha_{s}$ | $0.1179-0.0009$ | 0.1179 | $0.1179+0.0009$ | Diff.
---|---|---|---|---
$G_{\scriptscriptstyle{Z}}(\mu\mu),~\,\mathrm{MeV}$ | 83.985 | 83.985 | 83.984 | 0.001
$\Gamma_{\scriptscriptstyle{Z}},~\,\mathrm{MeV}$ | 2494.338 | 2494.814 | 2495.290 | 0.952
$G_{\scriptscriptstyle{W}}(l\nu),~\,\mathrm{MeV}$ | 678.995 | 678.981 | 678.967 | 0.028
$\Gamma_{\scriptscriptstyle{W}},~\,\mathrm{MeV}$ | 2089.607 | 2089.967 | 2090.326 | 0.719
$\sin^{2}\vartheta^{l}_{eff}\times 10^{6}$ | 231497 | 231500 | 231503 | 6
Table 6: The effect of the parametric uncertainty in $\alpha_{s}$ on the
magnitudes of pseudo-observables.

As seen from Tables 3 $-$ 5, the largest uncertainty comes to
$\Gamma_{\scriptscriptstyle{W}}$ and $\Gamma_{\scriptscriptstyle{Z}}$ due to
errors in $M_{\scriptscriptstyle{Z}}$. These parametric uncertainties remain,
however, well below the corresponding experimental errors.

###  3.2 Impact of new options

Numerical results for the comparison of versions are conveniently presented as
difference in values for a given observable at different sets of flags IHVP
and IAMT4.

####  3.2.1 Partial $G_{ij}$ and total $\Gamma_{tot}$ decay widths of the
Z-boson

IHVP, IAMT4 | 1,8 | 5,6 | 5,8 | $|\delta_{(1,8)-(5,8)}|$ | $|\delta_{(5,6)-(5,8)}|$
---|---|---|---|---|---
channel |  |  |  |  |
$G_{\nu,\bar{\nu}},~\,\mathrm{MeV}$ | 167.202 | 167.202 | 167.202 | 0 | 0
$G_{e^{+},e^{-}},~\,\mathrm{MeV}$ | 83.977 | 83.984 | 83.985 | 0.008 | 0.001
$G_{\mu^{+},\mu^{-}},~\,\mathrm{MeV}$ | 83.977 | 83.983 | 83.985 | 0.008 | 0.002
$G_{\tau^{+},\tau^{-}},~\,\mathrm{MeV}$ | 83.787 | 83.794 | 83.795 | 0.008 | 0.001
$G_{u,\bar{u}},~\,\mathrm{MeV}$ | 299.832 | 299.902 | 299.918 | 0.086 | 0.016
$G_{d,\bar{d}},~\,\mathrm{MeV}$ | 382.783 | 382.846 | 382.861 | 0.078 | 0.015
$G_{c,\bar{c}},~\,\mathrm{MeV}$ | 299.766 | 299.836 | 299.852 | 0.086 | 0.016
$G_{s,\bar{s}},~\,\mathrm{MeV}$ | 382.783 | 382.846 | 382.861 | 0.078 | 0.015
$G_{b,\bar{b}},~\,\mathrm{MeV}$ | 375.874 | 375.839 | 375.951 | 0.077 | 0.112
$G_{hadron},~\,\mathrm{MeV}$ | 1741.039 | 1741.268 | 1741.442 | 0.403 | 0.174
$\Gamma_{Z},~\,\mathrm{MeV}$ | 2494.387 | 2494.636 | 2494.814 | 0.427 | 0.178
Table 7:  Partial $G_{ij}$ and total $\Gamma_{Z}$ decay widths of the Z-boson
for sets of flags (IHVP, IAMT4): (1,8) in comparison with (5,6) and (1,8) in
comparison with (5,8).

In Table 7 we present the relevant numbers obtained with the latest update of
DIZET options and previous actual options of these flags, i.e. (IHVP, IAMT4):
(1,8) in comparison with (5,6) and (1,8) in comparison with (5,8). The main
improvement comes from accounting previously missing bosonic $O(\alpha^{2})$
corrections to the $Z\to b\bar{b}$ decay [16].

####  3.2.2 The effective weak mixing angle $\sin^{2}\vartheta_{eff}$

In Table 8 we illustrate various options for flag AMT4 in DIZET v. 6.45 to
estimate $\sin^{2}\vartheta_{eff}$ in different channels.

(IHVP, IAMT4) | (1,8) | (5,6) | (5,8) | $|\delta_{(1,8)-(5,8)}|\cdot 10^{3}$ | $|\delta_{(5,6)-(5,8)}|\cdot 10^{3}$
---|---|---|---|---|---
$\sin^{2}\vartheta_{eff}$ channel |  |  |  |  |
$\nu,\bar{\nu}$ | 0.231280 | 0.231149 | 0.231118 | 0.162 | 0.031
$e^{+},e^{-}$ | 0.231661 | 0.231530 | 0.231500 | 0.161 | 0.030
$\mu^{+},\mu^{-}$ | 0.231661 | 0.231530 | 0.231500 | 0.161 | 0.030
$\tau^{+},\tau^{-}$ | 0.231661 | 0.231530 | 0.231500 | 0.161 | 0.030
$u,\bar{u}$ | 0.231555 | 0.231424 | 0.231393 | 0.162 | 0.031
$d,\bar{d}$ | 0.231428 | 0.231297 | 0.231266 | 0.162 | 0.031
$c,\bar{c}$ | 0.231555 | 0.231424 | 0.231393 | 0.162 | 0.031
$s,\bar{s}$ | 0.231428 | 0.231297 | 0.231266 | 0.162 | 0.031
$b,\bar{b}$ | 0.232895 | 0.232970 | 0.232732 | 0.163 | 0.238
Table 8:  The effective weak mixing angle $\sin^{2}\vartheta_{eff}$ for all
channels calculated for sets of flags (IHVP, IAMT4): (1,8) in comparison with
(5,6) and (1,8) in comparison with (5,8).

The results for the channels are similar. The largest shift comes from
accounting previously missing bosonic $O(\alpha^{2})$ corrections to
$\sin^{2}\theta^{b}_{eff}$ [16].

####  3.2.3 Cross Sections

Figure 1: Differences $\delta$ defined in Eq. (3.2.3) for cross sections of
$e^{+}e^{-}\to u\bar{u}$ for sets of flags (IHVP, IAMT4): (1,8), (4,8), (5,6),
(5,7) relative to the current best set (5,8).

Using the example of the channel $e^{+}e^{-}\to u\bar{u}$, Figure 1 shows the
cross section differences for sets of the flags (IHVP, IAMT) relative to the
current best set (IHVP, IAMT) =(5,8);

$\delta=\displaystyle{\frac{\sigma(\rm{IHVP,IAMT})-\sigma(5,8)}{\sigma(5,8)}}\cdot
1000[\permil].$

The main influence on the result is the use of a modern parametrization for
the hadronic vacuum polarization. The effect of bosonic corrections is weaker.
The results close to the $Z$ peak ($\pm$ 1 GeV) show that the relative shift
is below $5\cdot 10^{-5}$.

####  3.2.4  Left-Right and Forward-Backward Asymmetries

The channel $e^{+}e^{-}\to u\bar{u}$ is used to show in Figures 2 and 3 the
differences for the left-right asymmetries and for the forward-backward
asymmetries for sets of the flags (IHVP, IAMT) compared with the current best
set (IHVP, IAMT) = (5,8);

$\Delta\rm{A}=\rm{A}(\rm{IHVP,IAMT})-\rm{A}(5,8).$

Figure 2: Differences for left-right asymmetries of $e^{+}e^{-}\to u\bar{u}$
for sets of flags (IHVP, IAMT4): (1,8), (4,8), (5,6), (5,7) from the current
best set (5,8). Figure 3: Differences for forward-backward asymmetries of
$e^{+}e^{-}\to u\bar{u}$ for sets of flags (IHVP, IAMT4): (1,8), (4,8), (5,6),
(5,7) from the current best set (5,8).

As in the case of corrections to the cross section the main effect is due to
changes in the hadronic vacuum polarization treatment. Bosonic corrections of
the $O(\alpha^{2})$ order taken into account change the left-right asymmetry
by no more than $3\cdot 10^{-4}$ and the forward-backward asymmetry by no more
than $2\cdot 10^{-4}$.

##  4 Benchmarks DIZET v. 6.45

Here, we provide a benchmark for the default set of parameters and flags.

     DIZET flags, see routine DIZET for explanation:
      IHVP = 5 IAMT4 = 8
      Iqcd = 3 Imoms = 1
     Imass = 0 Iscre = 0
     Ialem = 3 Imask = 0
     Iscal = 0 Ibarb = 2
     IFtjr = 1 Ifacr = 0
     IFact = 0 Ihigs = 0
     Iafmt = 3 Iewlc = 1
     Iczak = 1 Ihig2 = 0
     Iale2 = 3 Igfer = 2
     Iddzz = 1 Iamw2 = 0
     Isfsr = 1 Idmww = 0
     Idsww = 0
        IBAIKOV =          2012
        IBAIKOV =          2012

     DIZET input parameters:
     ZMASS   91.187600000000003      TMASS    172.75999999999999
     HMASS   125.25000000000000      WMASS    0.0000000000000000
     DAL5H   0.0000000000000000      ALQED5   137.03599908400000
     ALFAS   0.11790000000000000

     DIZET results:
     SIN2TW     0.22340388419691781
     WMASSsin   80.358790700232220
     WMASS      80.358790700232220
     DAL5H      2.7576193213462830E-002
     ALQED5     128.95030472145015
     ALST       0.10755034917841029
     ALPAS      0.11790000000000000

     CHANNEL         WIDTH         RHO_F_R        RHO_F_T        SIN2_EFF
     -------        -------       --------       --------       --------
     nu,nubar       167.202       1.007963       1.007963       0.231118
     e+,e-           83.985       1.005219       1.005062       0.231500
     mu+,mu-         83.985       1.005219       1.005062       0.231500
     tau+,tau-       83.795       1.005219       1.005062       0.231500
     u,ubar         299.918       1.005812       1.005757       0.231393
     d,dbar         382.861       1.006733       1.006723       0.231266
     c,cbar         299.852       1.005812       1.005757       0.231393
     s,sbar         382.861       1.006733       1.006723       0.231266
     t,tbar           0.000       0.000000       0.000000       0.000000
     b,bbar         375.839       0.994198       0.994198       0.232732
     hadron        1741.442
     total         2494.814

     W-widths
     lept,nubar     678.981
     down,ubar     1410.986
     total         2089.967

     FF:
     RHO             (0.99876990486335659,-4.73600008701756652E-003)
     RHO      (0.99876990  -.00473600)
     KAPPA.I  (1.03606482  0.01353154)
     KAPPA.J  (1.04380408  0.01353154)
     KAPPA.IJ (1.08147655  0.02706307)
     AL_I(s)                 (129.37048577139927,1.9810219926255057)
     AL_5_I(s)               (129.36196765366330,1.9807610678870036)
     ****************************************************

##  5 Conclusions

The new version 6.45 of the DIZET electroweak library is described. In this
work, we benchmark the novel implementation of two-loop $\alpha^{2}$ bosonic
and fermionic radiative corrections and several fits of the hadronic vacuum
polarization $\Delta\alpha^{(5)}_{had}(M_{\scriptscriptstyle{Z}})$ using
public versions of the AlphaQED code. The presented numerical results show the
impact of the new options for cross sections, left-right and forward-backward
asymmetries. We can see that the updates are relevant for high-precision
experiments at future electron-positron colliders. Compatibility with previous
versions of the code is supported. Predictions for observables and
pseudobservables have been produced with the ZFITTER program [2, 4]. The code
is available directly with the link: DIZET6.45. Recently the new electroweak
library GRIFFIN was created [66] in which comparisons with DIZET are given.
The new version of the discussed here DIZET program can serve for further
tuned comparisons in future high-precision studies.

Note: The work done in this paper is motivated by the HL-LHC studies and the
LHC electroweak Working Group activity and is based on the implementation of
the new version of the DIZET code released on December 2019 [14] and public
presentations of results during EWG meetings in 2019 [12, 68] and 2020 [70].

## Acknowledgments

This work has been supported in part by the Polish National Science Center
(NCN) under grant 2017/25/B/ST2/01987. A.A., L.K. and V.Ye. are grateful for
the support to RFBR grant N 20-02-00441.

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# Do People Engage in Motivated Reasoning to Think the World Is a Good Place
for Others?††thanks: This paper was previously circulated with the title: “The
Limits of Motivated Reasoning When Self-Image Is Not at Stake.” I would like
to thank Alberto Alesina, Roland Benabou, Christine Exley, David Laibson,
Matthew Rabin, and numerous seminar participants for helpful comments. I am
grateful for funding support from the Harvard Business School Research
Fellowship and the Eric M. Mindich Research Fund for the Foundations of Human
Behavior.

Michael Thaler Princeton University. Email<EMAIL_ADDRESS>
(October 2020)

###### Abstract

Motivated reasoning is a bias in inference in which people distort their
updating process in the direction of more attractive beliefs. Prior work has
shown how motivated reasoning leads people to form overly “positive” beliefs
that also serve to bolster one’s self-image in domains such as intelligence,
prosociality, and politics. In this paper, I study whether positivity-
motivated reasoning persists in domains where self-image does not play a role.
In particular, I analyze whether individuals motivatedly reason to think that
the world is a better place for others. Building off of the design from Thaler
(2020), I conduct a large online experiment to test for positivity-motivated
reasoning on issues such as cancer survival rates, others’ happiness, and
infant mortality. I find no systematic evidence for positivity-motivated or
negativity-motivated reasoning, and can rule out modest effects. Positivity is
not a sufficient condition for motivated reasoning.

##  1 Introduction

There is a common intuition in economics that people find it more attractive
to believe that they are in a “good” state of the world than in a “bad” state
of the world, and that this can lead them to form beliefs that are
directionally distorted in favor of good states. However, tests of such over-
optimism focus on states where self-image is at stake, such as about one’s
future prospects, one’s ability, one’s altruism, or one’s politics (e.g. [19];
[16]; [9]; [17]).

This paper argues that motivated reasoning — the distortion of new information
in the direction of more attractive beliefs — is not solely about “good” and
“bad” states. I run an experiment to explore how people make inferences about
states of the world that are good or bad for others, and in which self-image
does not play a role: positivity-motivated reasoning. I find evidence that
this form of positivity-motivated reasoning does not play much of a role in
inference.

In a large online experiment, I test whether people engage in positivity-
motivated reasoning on five topics: the survival rate of children with
leukemia, global poverty rates, annual deaths in armed conflict, others’
happiness levels, and infant mortality rates. The topics are in Table 1 below.

Topic | Positive Motives | Negative Motives
---|---|---
Infant mortality | Low / Decreasing | High / Increasing
Others’ reported happiness | High / Increasing | Low / Decreasing
Leukemia survival rate for children | High / Increasing | Low / Decreasing
Global poverty rate | Low / Decreasing | High /Increasing
Deaths in armed conflicts | Low / Decreasing | High / Increasing
Latitude of US | Neutral | Neutral
Table 1: The list of topics and positivity motives in the experiment. On the
computer, each topic is a hyperlink that links to the exact question wording
in Appendix B.

To identify motivated reasoning, I conduct a large online experiment that
builds off of the design of [17]. That paper found evidence of motivated
reasoning in political and performance domains. Forms of motivated reasoning
have also been found in other self-image domains such as about one’s altruism
([10]; [8]), intelligence ([16]; [9]), and attractiveness ([9]). Similar
theoretical work looks at ego-motivated beliefs, identity-motivated beliefs,
and other beliefs about the outside world (e.g. [12]; [3]; [5]; [2]). In
related experimental work, [1] does not find evidence for motivated reasoning
in domains that involve experimental payoffs but are not ego-relevant.

The main result in this paper is that – across several settings in which self-
image is not relevant – there is no evidence for positivity- or negativity-
motivated reasoning. I show that, aggregating across these questions, even
modest effects can be ruled out. In fact, by comparing the magnitude of
positivity-motivated reasoning to results in [17], we can rule out an effect
of positivity or negativity that is half as large as politically-driven or
performance-driven motivated reasoning. This evidence shows that positivity,
by itself, is insufficient for motivated reasoning.

The second result is that there is no evidence that subjects’ current beliefs
are reflective of past positivity-motivated reasoning. That is, subjects whose
beliefs are overly positive are no more likely to engage in positivity-
motivated reasoning in this experiment. This suggests that there is limited
heterogeneity in subjects’ positivity-motivated reasoning. Relatedly, there
are not substantial differences in motivated reasoning by demographic factors
like gender, education, or income.

The third result is that people do not expect to see evidence for positivity-
motivated reasoning, but do expect positivity to affect happiness. In a
separate survey, I ask participants what they expect the direction of
motivated reasoning to be about positivity, politics, and own performance.
While the majority of participants expect others to engage in pro-party and
pro-performance motivated reasoning, they are similarly likely to expect to
see positivity-motivated reasoning, negativity-motivated reasoning, or no
notable difference. Yet a clear majority of participants expect positive news
to make people happier.

Taken together, these results are consistent with the notion that motivated
reasoning is not only driven by belief-based utility. Subjects may attain
higher utility by learning that the world is good for other people, and yet
not systematically distort their inference process in favor of these beliefs.
That is, the beliefs that people find more attractive do not necessarily make
them happier. Rather, it may be limited to belief-based utility that relates
to one’s self-image.

The rest of the paper proceeds as follows: Section 2 discusses the main theory
and experimental design that identifies motivated reasoning, adapted from
[17]. Section 3 discusses the data. Section 4 presents the main experimental
results. Section 5 discusses interpretations of the main experiment and
presents survey evidence about what people expect about others’ behavior and
utility. Section 6 concludes and proposes directions for future work. The
appendices provide a table that is omitted from the main text, and list the
exact questions and pages that subjects see.

##  2 Theory and Experimental Design

###  2.1 Theory and Predictions

The theory of motivated reasoning follows [17]. Further details are in that
paper. When a motivated-reasoning agent infers about the probability that an
event is true ($T$) or false ($\neg T$), with prior $\mathbb{P}(T)$, the agent
forms his posterior by incorporating prior, likelihood, and a motivated
beliefs term:

$\underbrace{\mathbb{P}(T|x)}_{\text{posterior}}\propto\underbrace{\mathbb{P}(T)}_{\text{prior}}\cdot\underbrace{\mathbb{P}(x|T)}_{\text{likelihood}}\cdot\underbrace{M(T)^{\varphi(x)}}_{\text{mot.
reasoning}},$

We take log odds ratios to attain the additive form:

$\text{logit }\mathbb{P}(T|x)=\text{logit
}\mathbb{P}(T)+\log\left(\frac{\mathbb{P}(x|T)}{\mathbb{P}(x|\neg
T)}\right)+\varphi(x)(m(T)-m(\neg T)).$ (1)

The motivated reasoner acts as if he receives both the actual signal ($x$) and
a signal whose relative likelihood corresponds to how much he is motivated to
believe the state is $T$. $m(T):\\{T,\neg T\\}\to\mathbb{R}$ is denoted the
motive function. The weight put on this signal is $\varphi(x)\geq 0$, called
susceptibility. When $\varphi(x)=0$, the agent is Bayesian; when
$\varphi(x)>0$, the agent motivatedly reasons.

This paper will assume $\varphi(x)>0$ for the particular information structure
in the experiment below, and back out $m(T)-m(\neg T)$ from the inference
process. We will be interested in the psychology of the motive function. In
this paper, either $T$ will correspond to positivity (the world being a better
place) and $\neg T$ to negativity (the world being a worse place), or vice
versa.

The experiment provides people with not-obviously-uninformative signals about
the veracity of news sources. To fix ideas, consider the following question,
taken verbatim from the experiment:

[1cm]1cm Acute Myeloid Leukemia (AML) is a devastating illness in which
cancerous cells emerge in the bone marrow, invade the blood stream, and may
spread to the rest of the body. Tragically, hundreds to thousands of children
under the age of 15 are diagnosed with AML each year; it is one of the most
common cancers among children.

Of children under the age of 15 who are diagnosed with AML, what percent
survive for at least 5 years?

This is a question for which higher-valued states are more positive. The main
test of motivated reasoning then involves three steps:

  1. 1.

Beliefs: Subjects are asked to guess the answers to questions like the one
above. Importantly, they are asked and incentivized to guess their median
belief (i.e. such that they find it equally likely for the answer to be above
or below their guess).

  2. 2.

News: Subjects receive a binary message from one of two randomly-chosen news
sources: True News and Fake News. The message from True News is always
correct, and the message from Fake News is always incorrect. This is the main
(within-subject) treatment variation.

The message says either “The answer is greater than your previous guess of
[previous guess].” or “The answer is less than your previous guess of
[previous guess].” Note that the exact messages are different for each subject
since subjects have different guesses. These customized messages are designed
so that they have the same subjective likelihood of occurring.

For the cancer question above, “greater than” corresponds to Positive news and
“less than” to Negative news.

  3. 3.

Assessment: After receiving the message, subjects assess the probability that
the message came from True News using a scale from 0/10 to 10/10, and are
incentivized to state their true belief. This news veracity assessment is the
main outcome measure. The effect of variation in news direction on veracity
assessments is the primary focus for much of this paper.

More formally, consider an agent with prior $F(\theta)$ about a state in
$\Theta$. Denote by $\mu\equiv F^{-1}(1/2)$ the median of $F(\theta)$. For
simplicity, we assume that $F$ has no atom at $\mu$ and that
$\mathbb{P}(\mu=\theta)=0$. That is, the agent believes that the answer has
probability zero of being exactly equal to $\mu$, and the true probability is
indeed zero.111In this experiment, zero answers are correct, so the assumption
appears reasonable.

The agent receives a source that is either from True News ($T$) or Fake News
($\neg T)$. Both report one of two binary messages $G$ or $L$: “The answer
$\theta$ is greater than your median $\mu$” or “The answer $\theta$ is less
than your median $\mu$.” Prior beliefs $P(T)$ are fixed, and
$\log\left(\frac{\mathbb{P}(G|T)}{\mathbb{P}(G|\neg
T)}\right)=\log\left(\frac{\mathbb{P}(L|T)}{\mathbb{P}(L|\neg T)}\right)=0$ by
definition of a median.

| $\theta>\mu$ | $\theta<\mu$
---|---|---
True News sends | $G$ | $L$
Fake News sends | $L$ | $G$

The agent has a prior about the news source $p\equiv\mathbb{P}(T)$ that does
not depend on $\theta$, and infers about $\mathbb{P}(T)$ given the message
received.

We can now look at how a motivated reasoner updates his beliefs about the news
source after receiving $G$:

$\displaystyle\text{logit }P(T|G)$ $\displaystyle=\text{logit
}\mathbb{P}(T)+\log\left(\frac{\mathbb{P}(G|T)}{\mathbb{P}(G|\neg
T)}\right)+\varphi\left(m(\theta|\theta>\mu)-m(\theta|\theta<\mu)\right)$
$\displaystyle=\text{logit
}p+\varphi\left(m(\theta|\theta>\mu)-m(\theta|\theta<\mu)\right).$

Therefore, if $m$ is strictly monotonically increasing in $\theta$, then
$P(T|G)>P(T|L)$, and if $m$ is strictly monotonically decreasing in $\theta$,
then $P(T|G)<P(T|L)$. By contraposition, if $P(T|G)=P(T|L)$, then $m$ is
neither strictly monotonically increasing nor decreasing in $\theta$.

Additionally, if $m$ is monotonic in $\theta$ for all agents but there is
heterogeneity in its slope, then the average slope may be zero because some
agents have upward-sloping motives (“positivity motives”) and some agents have
downward-sloping motives (“negativity motives”). In this case, if agents have
received information drawn from the same distribution in the past, then their
current beliefs will reflect their motives. A positivity-motivated reasoner
will be more likely to hold a belief that $\mu>\theta$, and a negativity-
motivated reasoner with a decreasing motive function will be more likely to
hold a belief that $\mu<\theta$. This implies that an agent who believes
$\mu>\theta$ is more likely to believe that $P(T|G)>P(T|L)$ in the experiment,
and an agent who believes $\mu<\theta$ is more likely to believe that
$P(T|G)<P(T|L)$ in the experiment.

That is, if motive direction is heterogeneous, subjects will trust Fake News
more than True News. For further details, see [17]. By contraposition, if
subjects trust Fake News and True News equally, then there is no evidence for
heterogeneity in positivity- versus negativity-motivated reasoning.

###  2.2 Experimental Details

The experiment follows the structure and incentive scheme of [17], which
contains further details. Screenshots of a version of each page in the
experiment, including instructions and scoring rules, can be found in Appendix
C.

Subjects first see an Introduction page for consent, then a Demographics page,
and then the instructions and point system for Question pages. On each
Question page, subjects are asked and incentivized to give a median guess, a
lower bound (equal to their 25th-percentile belief), and an upper bound (equal
to their 75th-percentile belief). The median is incentivized using a linear
scoring rule, and the bounds using piecewise-linear scoring rules. For
details, see Appendix C.

Next, subjects see the instructions and point system for News pages. On each
News page, subjects see the message that says whether the answer is greater
than or less than their previous median guess, and are asked and incentivized
to assess the probability that the message comes from True News versus Fake
News using a quadratic loss scoring rule. Subjects are told that the ex ante
probability of True News is 1/2. They are also asked to give an updated median
guess after seeing the message, and are again incentivized with a linear
scoring rule. For details, see Appendix C.

Subjects see News pages after their corresponding Question page, in the order:
Question 1, News 1, Question 2, News 2, …. At the end of the experiment, they
see a Results page with details on all the correct answers, points scored, and
money earned.

At the end of the experiment, subjects’ points earned on each part of the
experiment are averaged. Subjects are paid a $3 show-up fee and have a
probability of winning a $10 bonus equal to their average score divided by
1000. This probabilistic bonus is designed to eliminate potential hedging and
risk-aversion confounds.

##  3 Data

The experiment was conducted on Amazon’s Mechanical Turk (MTurk) platform.
MTurk is an online labor marketplace in which participants choose “Human
Intelligence Tasks” to complete. MTurk has become a very popular way to run
economic experiments (e.g. [11]; [13]), and [14] find that participants
generally tend to have more diverse demographics than students in university
laboratories on dimensions like age and politics. The experiment was coded
using oTree, an open-source software based on the Django web application
framework developed by [6].

Wave 1 was conducted on July 8-9, 2019, and asked about the leukemia survival
rates question. Wave 1 additionally included political and performance
questions that were part of a separate experiment. Wave 2 was conducted on
October 1-2, 2019, and asked about the other four questions. Both waves were
offered to MTurk workers currently living in the United States who had not
previously taken one of my motivated reasoning experiments. 522 participants
from Wave 1 and 508 participants from Wave 2 passed simple attention and
comprehension checks.222In order to pass these checks, subjects needed to
perfectly answer the comprehension check question in Appendix B (by giving a
correct answer, correct bounds, and answering the news assessment with
certainty). In addition, many questions had clear maximum and minimum possible
answers (such as percentages, between 0 and 100). Subjects were dropped if any
of their answers did not lie within these bounds. Wave 1 also included
politicized questions and tested a debiasing treatment that is unrelated to
this paper; only participants in the control group are included here, and only
observations on the positivity questions are kept.

Subjects in Wave 1 answer one question about positivity; subjects in Wave 2
answer four questions about positivity and one neutral question. There are a
total of 3,062 guesses to these questions. Zero guesses were exactly
correct.333This suggests, reassuringly, that subjects did not look up the
correct answers. There are therefore a total of 3,062 news assessments. 2,554
assessments are Positive or Negative, and 508 assessments are on the neutral
topic.

The balance table for the Positive / Negative treatment is in Section A.1.
Since this randomization is within subject, treatments are expected to be
balanced across demographics. The overall shares of Positive and Negative are
not statistically significantly different, indicating that there is not
substantially different attrition.

##  4 Results

###  4.1 Raw Data

This subsection shows that the raw data does not support positivity- or
negativity-motivated reasoning, and the following subsection shows the
relevant regressions.

The mean assessment of Positive news is 57.7 percent (s.e. 0.7 percent) and
the mean assessment of Negative news is 58.5 percent (s.e. 0.8 percent).444All
standard errors are clustered at the individual level. The difference between
these is -0.7 percentage points; this point estimate is statistically
insignificantly different from zero ($p=0.457$).

Figure 2 shows the empirical distributions of assessments for Positive and
Negative news, which substantially overlap.

Likewise, there is no evidence that current beliefs are reflective of
motivated beliefs on these questions. The mean assessment of Error-
Accentuating news is 58.0 percent (s.e. 0.7 percent) and the mean assessment
of Error-Mitigating news is 58.2 percent (s.e. 0.8 percent). The difference
between these is -0.3 percentage points; this point estimate is statistically
insignificantly different from zero ($p=0.800$).

Figure 2 shows the empirical distributions of assessments for True News and
Fake News, which substantially overlap.

Figure 1: Histogram of Perceived Veracity of Positive and Negative News

Figure 2: Histogram of Perceived Veracity of True and Fake News About
Positivity

  * •

Notes: Only Positive / Negative news observations, as defined in Table 1.
Messages are customized so that Bayesians give the same assessment for
Positive and Negative news.

###  4.2 Main Specifications

Due to subjects in the two waves seeing different questions, the main
specification is between subjects. In particular, subjects in Wave 1 only see
one positivity-related question, so the within-subject test essentially
ignores this sample.

In particular, the main specification for positivity-motivated reasoning is in
Section 4.2, column 1. The regression looks at assessments $a$ for subject
$i$, question topic $q$, and round $r$ with fixed effects for $q$ and $r$ when
all news is Positive or Negative:

$a_{iqr}=\alpha+\beta\cdot 1(\text{Pro-Party})_{iqr}+\gamma z_{i}+\delta
FE_{q}+\zeta FE_{r}+\epsilon_{iqr}$

$z_{i}$ is a vector of controls. The controls used are age, an indicator for
political party, an indicator for race, an indicator for gender, log(income),
years of education, and an indicator for whether the subject is part of a
religious group.

Column 2 uses the within-subject design; the standard error is larger, but the
coefficient does not substantially change. In order to test whether there are
differences compared to Neutral news, column 3 includes indicators for both
Positive (versus Neutral) news and Negative (versus Neutral) news.

Columns 4 and 5 regress assessments on an indicator for True News (as opposed
to Fake News), with and without controls for positivity. Recall from Section 2
that this measures whether directional errors in current beliefs are partly
explained by past motivated reasoning on these topics, and therefore whether
we should expect much heterogeneity in motive direction.

[!htb] The Effect of News Direction and Actual Veracity on Perceived Veracity
(1) (2) (3) (4) (5) Positive News -0.005 -0.017 0.007 -0.005 (0.010) (0.016)
(0.016) (0.010) Negative News 0.012 (0.016) True News -0.002 0.000 (0.010)
(0.011) Neutral News No No Yes No No Question FE Yes Yes Yes Yes Yes Round FE
Yes Yes Yes Yes Yes Subject controls Yes No Yes Yes Yes Subject FE No Yes No
No No Observations 2554 2554 3062 2554 2554 $R^{2}$ 0.03 0.49 0.03 0.03 0.03
Mean 0.581 0.581 0.577 0.581 0.581 Standard errors in parentheses ∗ $p<0.10$,
∗∗ $p<0.05$, ∗∗∗ $p<0.01$

  * •

Notes: OLS, errors clustered at subject level. Neutral News indicates that
Positive / Negative news assessments are compared to assessments on Neutral
topics. These classifications are defined in Table 1. Controls: age, political
party, race, gender, log(income), years of education, and member of religious
group.

Every single coefficient in Section 4.2 is insignificant and each point
estimate is within 2 pp of zero. Modest effect sizes can be ruled out at the
95-percent significance level. There is no evidence for aggregate-level
positivity-motivated reasoning or negativity-motivated reasoning. There is no
evidence that subjects infer differently on positivity-related topics compared
to the neutral topic. There is no evidence that subjects have formed erroneous
beliefs in the direction of their motivated reasoning.

An alternative measure of motivated reasoning, which looks at subjects update
their beliefs about the original question, generates the same prediction.
After seeing the message, subjects’ updated median belief is elicited. 38
percent of the time, subjects update in the positive direction (s.e. 1
percent). 38 percent of the time, subjects update in the negative direction
(s.e. 1 percent). 24 percent of the time, subjects stay with their original
guess (s.e. 1 percent). Clearly, there is no systematic updating in the
positive or negative direction.

###  4.3 Heterogeneity

The results above show that there is no aggregate evidence for positivity- or
negativity-motivated reasoning. This may be because nobody engages in
motivated reasoning or because there is some heterogeneity with mean zero.
This section discusses two forms of heterogeneity: heterogeneity across
people, and heterogeneity across questions.

As shown in columns 4 and 5 of Section 4.2, there is no evidence that subjects
have formed their current beliefs on these topics because of past motivated
reasoning. This suggests that the degree of heterogeneity across people is not
likely to be large. In support of this, Section 4.3 shows that treatment
effects are not especially heterogeneous across demographic groups in
systematic ways.

[htb!] Heterogeneity in Positivity-Motivated Reasoning: Horse Race Regression
(1) (2) (3) (4) (5) (6) (7) (8) Pos News x Male 0.03 0.02 (0.02) (0.02) Pos
News x (Age$>$32) -0.04∗ -0.03∗ (0.02) (0.02) Pos News x White -0.01 -0.01
(0.02) (0.02) Pos News x College -0.02 -0.01 (0.02) (0.02) Pos News x
(Inc$>$50K) -0.01 -0.01 (0.02) (0.02) Pos News x Democrat -0.03 -0.03 (0.02)
(0.02) Pos News x Republican 0.04 0.05∗ (0.03) (0.03) Pos News x Religious
-0.00 -0.01 (0.02) (0.02) Pos News Yes Yes Yes Yes Yes Yes Yes Yes Question FE
Yes Yes Yes Yes Yes Yes Yes Yes Round FE Yes Yes Yes Yes Yes Yes Yes Yes
Subject controls Yes Yes Yes Yes Yes Yes Yes Yes Observations 2554 2554 2554
2554 2554 2554 2554 2554 $R^{2}$ 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Mean
0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 Standard errors in parentheses ∗
$p<0.10$, ∗∗ $p<0.05$, ∗∗∗ $p<0.01$

  * •

Notes: OLS regression coefficients, errors clustered at subject level. FE
included for round number and topic. Only Positive / Negative news
observations, as defined in Table 1. Religious: subject affiliates with any
religion. Controls: age, political party, race, gender, log(income), years of
education, and religious.

The evidence is not precise enough to clearly argue in favor of or against
between-topic heterogeneity. On one question (regarding global poverty rates),
there is statistically significant evidence for negativity-motivated
reasoning: the difference between Negative and Positive news assessments is 11
pp (s.e. 3 pp).555There are several potential explanations for this. It may be
noise, and it may also be a question on which self-image does play a role. In
particular, people may engage in social comparison and be motivated to believe
that many others are less well-off. The treatment effect on news assessments
for the other questions is neither large nor significant at 95 percent
confidence levels.666The overall effect is still close to zero if we remove
the global poverty question. In the between-subject test, the coefficient on
Positive news is 0.019 (s.e. 0.11; $p=0.068$); in the within-subject test, the
coefficient is 0.009 (s.e. 0.19; $p=.647$). Effects of larger than 4 pp lie
outside the 95-percent confidence interval. One direction for future work is
to extend the domain of topics analyzed to determine whether the global
poverty question is an outlier or in a different motivated category such as
social comparisons.

##  5 Discussion

Results from the experiment indicate that there is no evidence for positivity-
or negativity-motivated reasoning. To better understand the precision of these
results, it is helpful to compare the results to those in [17]. Since the
experimental design is the same, the comparison has the same units. Treatment
effects for the three categories are plotted in Figure 3.

Figure 3: Comparing Motivated Reasoning About Positivity to Politics and
Performance

  * •

Notes: Treatment effects for the effect of Positive versus Negative, Pro-Party
versus Anti-Party, and Pro-Performance versus Anti-Performance on news
veracity assessments. Pro-Party and Pro-Performance coefficients from [17].
Error bars correspond to 95 percent confidence intervals.

It is easily apparent that the effect of Positive news is significantly
different than the effect of seeing Pro-Party news or Pro-Performance news.

There are two sets of explanations for the experimental results. First,
motives may be systematically different from belief-based utility, leading
people to distort how they process information differently in the self-image-
relevant domains from the positivity domains. Second, people may not actually
receive utility from holding beliefs from positivity.

As a suggestive test to separate these different hypotheses, I run two follow-
up surveys among a new group of subjects — drawn from different Mechanical
Turk samples — on January 8-9 and 13, 2020. There are 303 participants in
Survey 1 and 167 in Survey 2.777This number does not include 16 subjects in
Survey 1 and 5 subjects in Survey 2 who failed a simple attention check
question. Results do not qualitatively change with the inclusion of these
subjects. The evidence is consistent with the hypothesis that motives are
systematically different from belief-based utility, and that survey
participants are aware of this.

In Survey 1, participants were given a definition of motivated reasoning and
asked to predict the direction of motivated reasoning about positivity,
politics, and performance, and given sample topics on all three. On
positivity, they were asked whether they thought that most people motivatedly
reasoned in the direction of believing that the world was a better place for
others, most people motivatedly reasoned in the direction of believing the
world was a worse place for others, or about the same. The example topics were
infant mortality, happiness, and cancer survival rates.

The results from this survey are shown in Figure 4. 65 percent of subjects
expect motivated-reasoning distortions in the Pro-Party direction, versus only
16 percent who expect distortions in the Anti-Party direction. Similarly, 56
percent expect Pro-Performance distortions, and only 18 percent expect Anti-
Performance distortions. Subjects, however, were similarly likely to predict
distortions in the Positive (36 percent) and Negative (30 percent) directions;
this difference is not statistically significant ($p=0.231$). They were also
more likely to predict no directional distortions about positivity (34
percent) than about party (18 percent) or performance (26 percent). Each
answer about positivity is statistically significantly different from each
answer about party and performance at the 5-percent level.

This suggests that the experimental results — that motivated reasoning occurs
about politics and performance, but not directionally about positivity — are
anticipated by the sample as a whole.

In Survey 2, participants were given the same categories and examples, but now
asked to predict whether such beliefs would lead people to be happier. The
results are shown in Figure 5. Unlike with motivated reasoning, a clear
majority of 69 percent believes that positivity makes people happier, and only
10 percent believes that negativity increases happiness. Politics and
performance have similar and statistically indistinguishable point estimates.

Figure 4: People Systematically Expect Motivated Reasoning About Politics and
Performance, But Do Not About Positivity

  * •

Notes: The y-axis is the share of respondents who stated that they expect most
people to have motivatedly reason in one direction, the other direction, or a
similar amount in both directions. 0.4 percent of questions are left
unanswered, and are coded as “Similar.” Error bars correspond to 95 percent
confidence intervals. The differences between each of the Positivity bars and
their corresponding bars in the Party and Performance columns are
statistically significant at the 95 percent level.

Figure 5: People Expect Happiness Due to Beliefs About Positivity, Not Just
About Politics and Performance

  * •

Notes: The y-axis is the share of respondents who stated that they expect most
people to be happier when receiving news in one direction, the other
direction, or a similar amount in both directions. Error bars correspond to 95
percent confidence intervals. The differences between each of the Positivity
bars and their corresponding bars in the Party and Performance columns are not
statistically significant at the 95 percent level.

Taken as a whole, the evidence is in support of the explanation that people do
receive utility (in the form of happiness) from believing that the world is
good for others, but not in support of this form of positivity influencing
motives. That is, motives do not include other-regarding belief-based utility.
This explanation may also help explain action-induced motivated beliefs, where
people distort their beliefs as an excuse to not be generous to others ([10];
[8]).

##  6 Conclusion

This paper has shown that people do not necessarily motivatedly reason in the
direction of “good” states when their self-image is not at stake. When asked
about positive or negative news about others, this experiment finds no
evidence of systematic directional distortions of how people process the
information. It also does not indicate any evidence that people’s current
beliefs are distorted due to such positivity- or negativity-motivated
reasoning.

Survey results additionally show that people think that believing in positive
states of the world does not induce motivated reasoning. However, people do
think that believing in these positive states leads to increased happiness.
The results from this paper also suggest that utility-maximizing beliefs do
not necessarily explain why people form persistently inaccurate beliefs.

One direction for future work is to better understand the relationship between
the belief-based utility function and the motive function. For instance, do
particular emotions affect utility and motives differently? These results
suggest that happiness is insufficient for motives; however, in many self-
image domains, pride and identity confirmation play larger roles than
happiness. An alternative hypothesis in psychology, proposed by [18], is that
self-deception is a mechanism by which people can deceive others. Convincing
others that the world is a good place can be less impactful than convincing
others that one is smarter, more altruistic, or more correct.

Once the tether between the motive function and the utility function is
removed, future work can treat the motive function as a separate object for
study. The objective is for this experimental design to then be used to elicit
motives and analyze their role in other economic contexts.

## References

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##  Appendix A Supplementary Appendix: Additional Tables

###  A.1 Balance Table

| Negative News | Positive News | Neg vs. Pos |  p-value
---|---|---|---|---
Democrat | 0.479 | 0.468 | 0.011 | 0.567
| (0.014) | (0.014) | (0.020) |
Republican | 0.195 | 0.193 | 0.002 | 0.911
| (0.011) | (0.011) | (0.016) |
Male | 0.550 | 0.532 | 0.018 | 0.353
| (0.014) | (0.014) | (0.020) |
Female | 0.442 | 0.458 | -0.016 | 0.406
| (0.014) | (0.014) | (0.020) |
Age | 35.353 | 35.732 | -0.379 | 0.357
| (0.291) | (0.290) | (0.411) |
Education | 14.919 | 14.873 | 0.046 | 0.538
| (0.052) | (0.054) | (0.075) |
Log(Income) | 10.833 | 10.853 | -0.021 | 0.520
| (0.023) | (0.022) | (0.032) |
White | 0.745 | 0.730 | 0.015 | 0.374
| (0.012) | (0.012) | (0.017) |
Black | 0.090 | 0.088 | 0.002 | 0.881
| (0.008) | (0.008) | (0.011) |
Hispanic | 0.046 | 0.057 | -0.012 | 0.180
| (0.006) | (0.006) | (0.009) |
Asian | 0.087 | 0.090 | -0.003 | 0.789
| (0.008) | (0.008) | (0.011) |
Religious | 0.470 | 0.479 | -0.009 | 0.649
| (0.014) | (0.014) | (0.020) |
$N$ | 1248 | 1306 | 2554 |

Notes: Standard errors in parentheses. Education is in years. Religious is 1
if subject in any religious group. Positive and Negative News is defined in
Table 1.

##  Appendix B Study Materials: Exact Question Wordings

### Infant Mortality

The CDC provides statistics for mortality rates for infants. In 1997, there
were 28.0 thousand infant deaths in the United States.

How many thousands of infant deaths in the United States were there in 2017
(the most recent year available)?

(If you answer X, it means you think that there were X thousand deaths.)

Correct answer: 22.3.

Source linked on results page:
https://www.cdc.gov/nchs/data/nvsr/nvsr68/nvsr68_10-508.pdf

### Others’ Happiness

Many surveys ask the following question about subjective happiness:

“Please imagine a ladder with steps numbered from zero at the bottom to ten at
the top. Suppose we say that the top of the ladder represents the best
possible life for you and the bottom of the ladder represents the worst
possible life for you. If the top step is 10 and the bottom step is 0, on
which step of the ladder do you feel you personally stand at the present
time?”

In 2006, the average subjective happiness level in the United States was 7.18
out of 10.

What was the average subjective happiness level in the US in 2018?

Correct answer: 6.88.

Source linked on results page: https://ourworldindata.org/happiness-and-life-
satisfaction

### Cancer in Children

Acute Myeloid Leukemia (AML) is a devastating illness in which cancerous cells
emerge in the bone marrow, invade the blood stream, and may spread to the rest
of the body. Tragically, hundreds to thousands of children under the age of 15
are diagnosed with AML each year; it is one of the most common cancers among
children.

Of children under the age of 15 who are diagnosed with AML, what percent
survive for at least 5 years?

(Please guess between 0 and 100.)

Correct answer: 68.8.

Source linked on results page: https://www.lls.org/facts-and-
statistics/overview/childhood-blood-cancer-facts-and-statistics

### Global Poverty

Around the world, many people do not have enough money for basic necessities.
The World Bank defines extreme poverty as having less than the equivalent of
$1.90 per day.

In 1990, the World Bank estimated that 1897 million people around the world
were living in extreme poverty.

As of 2015 (the most recent year available), how many milllions of people
around the world live in extreme poverty?

(If you answer X, it means you think that X million people live in extreme
poverty.)

Correct answer: 731.

Source linked on results page: http://povertydata.worldbank.org/poverty/home/

### Armed Conflict

The Department of Peace and Conflict Research estimates that 45.8 thousand
people were killed per year in battles in the fifteen years from 1989-2003.

How many thousands of people were killed per year in battles in the fifteen
years from 2004-2018?

(If you answer X, it means you think that X thousand people were killed per
year.)

Correct answer: 48.12

Source linked on results page:
https://www.pcr.uu.se/digitalAssets/667/c_667494-l_1-k_battle-related-deaths-
by-region--1989-2018.pdf

### Latitude of Center of the United States

The U.S. National Geodetic Survey approximated the geographic center of the
continental United States. (This excludes Alaska and Hawaii, and U.S.
territories.)

How many degrees North is this geographic center?

(Please guess between 0 and 90. The continental U.S. lies in the Northern
Hemisphere, the Equator is 0 degrees North, and the North Pole is 90 degrees
North.)

Correct answer: 39.833.

Source linked on results page: http://bit.ly/center-of-the-us

### Comprehension Check: Current Year

In 1776 our fathers brought forth, upon this continent, a new nation,
conceived in Liberty, and dedicated to the proposition that all men are
created equal.

What is the year right now?

This is not a trick question and the first sentence is irrelevant; this is a
comprehension check to make sure you are paying attention. For this question,
your lower and upper bounds should be equal to your guess if you know what
year it currently is.

Correct answer: 2019.

Source linked on results page: http://bit.ly/what-year-is-it

##  Appendix C Online Appendix: Study Materials