This expression is $>0$ as can be seen by substituting $gv_{l}$ for $v_{h}$
and $1-b_{l}$ for $\Delta sp_{l}$:

$\displaystyle S((\Delta sp_{l}-\Delta sp_{h})S+v_{h}-\Delta sp_{l}v_{l})\geq$
(7) $\displaystyle S[(\Delta sp_{l}-\Delta sp_{h})S+gv_{l}-(1-b_{l})v_{l}]=$
$\displaystyle S[(\Delta sp_{l}-\Delta sp_{h})S+(g-1+b_{l})v_{l}]>0$

Thus, in contests where the performance gap between low and high types is
large enough such that $g\geq 1$ is satisfied, the expected gain from self-
promotion for a low type is larger than for a high type. $\blacksquare$

A high type will of course self-promote if the gain from self-promotion is
higher than its costs. More specifically:

###### Lemma 2

If, for a high type with value $v_{h}$ and a performance gap $g\geq 1$,
$c_{p}<\frac{\Delta sp_{h}(S-v_{h})}{S(S-\Delta sp_{h})}$ in equilibrium, then
all agents will engage in self-promotion.

##### Proof.

If the high type did not engage in self-promotion, her winning probability
would fall by $\frac{v_{h}}{S}-\frac{v_{h}-\Delta sp_{h}}{S-\Delta
sp_{h}}=\frac{\Delta sp_{h}(S-v_{h})}{S(S-\Delta sp_{h})}$. By assumption,
this is larger than $c_{p}$, so her net gain would be negative. She will
therefore adhere to self-promotion. By Lemma 1, this continues to hold for the
low types if $g\geq 1$. Alternatively, low types will engage in self-promotion
if $c_{p}<\frac{\Delta sp_{l}(S-v_{l})}{S(S-\Delta sp_{l})}$. $\blacksquare$

###  B.4 Sabotage

If an agent sabotages another agent, this decreases that agent’s output
$v_{i}$ and thus decreases $S$, which increases that agents probability of
winning the contest. This decrease in $S$ also benefits all other agents and
thus sabotage has—contrary to self-promotion—an important negative externality
(Konrad, 2000). We first show that if an agent sabotages another agent, the
agent sabotages all agents of that type.

###### Lemma 3

The expected marginal gain in the probability of winning increases with the
number of other agents (of the same type) being sabotaged.

##### Proof.

Consider a high type of value $v_{h}$ and sincere rating by everyone. Her
probability of winning is $\frac{v_{h}}{S}$. If she sabotages one more high
type, her probability of winning increases to $\frac{v_{h}}{S-b_{h}}$. The
marginal gain in probability from this act of sabotage is
$\frac{v_{h}}{S-b_{h}}-\frac{v_{h}}{S}=\frac{b_{h}v_{h}}{S(S-b_{h})}$. We can
then express the marginal gain of sabotage as a function of the number of
other high types $x_{h}$ that the agent sabotages:
$f(x_{h})=\frac{x_{h}b_{h}v_{h}}{S(S-x_{h}b_{h})}$. The first derivative of
which is $f^{\prime}(x_{h})=\frac{b_{h}v_{h}}{(S-x_{h}b_{h})^{2}}$ which is
always positive for positive $x_{h}$ and given that
$S>nb_{h}>hb_{h}>x_{h}b_{h}$ holds since by design $n>h$ and $h>x_{h}$ because
the agent does not sabotage herself and at most sabotages $h-1$ high types.
Thus, the marginal gain of sabotaging another high type increases with the
number of high types already sabotaged. If the costs of sabotage are smaller
than the (maximum) marginal gain from sabotaging the last other high type
agent, a rational high type agent will therefore sabotage all other high type
agents. An analogous argument shows that the marginal gain for a high type
from sabotaging one more low type agent increases with the number of low type
agents already sabotaged. The same arguments also hold for low types
sabotaging other agents. $\blacksquare$

We can use Lemma 3 to compute the bounds of sabotage costs $c_{s}$ when agents
engage in strategic behavior. For example, consider a high type of value
$v_{h}$ not currently sabotaging anyone and sincere voting by everyone else so
that $v_{h}=b_{h}(n+l+h)$ and $v_{l}=b_{l}(n+l+h)$, and $S=hv_{h}+lv_{l}$. If
she sabotages all other high types (remember she will not sabotage herself)
her probability of winning is $\frac{v_{h}}{S-b_{h}(h-1)}$. If she were to
sabotage only $h-2$ high types her probability of winning is
$\frac{v_{h}}{S-b_{h}(h-2)}$. Consequently, her maximal marginal increase in
the probability of winning from sabotaging other high types results from
sabotaging all other $h-1$ instead of $h-2$ high type agents and is of size

$\displaystyle\frac{v_{h}}{S-b_{h}(h-1)}-\frac{v_{h}}{S-b_{h}(h-2)}=\frac{v_{h}b_{h}}{(S-b_{h}(h-1))(S-b_{h}(h-2))}$
(8)

If the costs of sabotage $c_{s}$ are smaller than the maximal marginal gain in
Eq. 8, a rational high type will therefore sabotage all other high types. In a
similar fashion we can compute the bounds for all other acts of sabotage (high
types sabotaging low types, low types sabotaging high types etc.). We show all
bounds in the appendix. Inspecting the bounds, the following relationships
between the bounds are apparent.

###### Lemma 4

Both high and low types have more to gain from self-promotion than from
sabotaging any other types.

##### Proof.

This follows directly from the externality associated with sabotage.
$\blacksquare$

###  B.5 High Types Start Sabotaging High Types

###### Lemma 5

High types have more to gain from sabotaging other agents of a given type (all
low types or all high types), than low types have to gain from sabotaging
those agents.

##### Proof.

Consider a high type of value $v_{h}$ not currently sabotaging anyone and
sincere voting by everyone else so that $v_{h}=b_{h}(n+l+h)$ and
$v_{l}=b_{l}(n+l+h)$, and $S=hv_{h}+lv_{l}$. High types will sabotage other
high types if the cost of sabotage $c_{s}$ is lower than the maximum marginal
increase of sabotaging the last unit (i.e., going from sabotaging $h-2$ to
$h-1$).

$\displaystyle\frac{v_{h}}{S-b_{h}(h-1)}-\frac{v_{h}}{S-b_{h}(h-2)}$ (9)
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}(S-b_{h}(h-2))-v_{h}(S-b_{h}(h-1))}{(S-b_{h}(h-1))(S-b_{h}(h-2))}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}S-v_{h}b_{h}(h-2)-v_{h}S+v_{h}b_{h}(h-1)}{(S-b_{h}(h-1))(S-b_{h}(h-2))}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}b_{h}}{(S-b_{h}(h-1))(S-b_{h}(h-2))}$

$\blacksquare$

###  B.6 High Types Start Sabotaging Low Types

There are two possible transition points into this equilibrium: High types may
start sabotaging low types while high types already sabotage other high types,
but no sabotage otherwise. Or high types may start sabotaging low types, after
low types have already started to sabotage high types. That is, the exact
bound for high types to sabotage low types, will depend on whether it falls
before or after the bound of low types sabotaging high types (in terms of
$c_{s}$) as this determines the behavior of low types, which in turn affects
the costs of sabotaging behavior for high types. We compute both bounds and
then compare them to establish this sequence.

###### Lemma 6

Both high and low types have more to gain from sabotaging high types than low
types.

####  B.6.1 Option H1: High types sabotage all other high types and low types
do not sabotage anyone

Consider a high type of value $v_{h}$ not currently sabotaging anyone and
sincere voting by everyone else so that $v_{h}=b_{h}(n+l+h)$ and
$v_{l}=b_{l}(n+l+h)$, and $S=hv_{h}+lv_{l}$. In this equilibrium, $h$ high
types vote $-b_{h}$ on $h-1$ others.

$\displaystyle\frac{v_{h}}{S-hb_{h}(h-1)-b_{l}l}-\frac{v_{h}}{S-hb_{h}(h-1)-b_{l}(l-1)}$
(10) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}(S-hb_{h}(h-1)-b_{l}(l-1))-v_{h}(S-hb_{h}(h-1)-b_{l}l)}{(S-hb_{h}(h-1)-b_{l}l)(S-hb_{h}(h-1)-b_{l}(l-1))}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}S-v_{h}hb_{h}(h-1)-v_{h}b_{l}(l-1)-v_{h}S+v_{h}hb_{h}(h-1)+v_{h}b_{l}l}{(S-hb_{h}(h-1)-b_{l}l)(S-hb_{h}(h-1)-b_{l}(l-1))}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}b_{l}}{(S-hb_{h}(h-1)-b_{l}l)(S-hb_{h}(h-1)-b_{l}(l-1))}$

####  B.6.2 Option H2: High and low types sabotage all high types

Consider a high type of value $v_{h}$ not currently sabotaging anyone and
sincere voting by everyone else so that $v_{h}=b_{h}(n+l+h)$ and
$v_{l}=b_{l}(n+l+h)$, and $S=hv_{h}+lv_{l}$. In this equilibrium, $h$ high
types vote $-b_{h}$ on $h-1$ others and $l$ low types vote $-b_{h}$ on $h$
high types. This turns out to be the relevant bound (see proof below).

$\displaystyle\frac{v_{h}}{S-hb_{h}(h-1)-lb_{h}h-b_{l}l}-\frac{v_{h}}{S-hb_{h}(h-1)-lb_{h}h-b_{l}(l-1)}$
(11) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}(S-hb_{h}(h-1)-lb_{h}h-b_{l}(l-1))-v_{h}(S-hb_{h}(h-1)-lb_{h}h-b_{l}l)}{(S-hb_{h}(h-1)-lb_{h}h-b_{l}l)(S-hb_{h}(h-1)-lb_{h}h-b_{l}l)}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}S-v_{h}hb_{h}(h-1)-v_{h}lb_{h}h-v_{h}b_{l}(l-1)-v_{h}S+v_{h}hb_{h}(h-1)+v_{h}lb_{h}h+v_{h}b_{l}l}{(S-hb_{h}(h-1)-lb_{h}h-b_{l}l)(S-hb_{h}(h-1)-lb_{h}h-b_{l}l)}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}b_{l}}{(S-hb_{h}(h-1)-lb_{h}h-b_{l}l)(S-hb_{h}(h-1)-lb_{h}h-b_{l}l)}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{h}b_{l}}{(S-hb_{h}(h-l-1)-b_{l}l)(S-hb_{h}(h-l-1)-b_{l}l)}$

$\blacksquare$

###  B.7 Low Types Start Sabotaging High Types

Similarly, there are two possible bounds when low types start to sabotage high
types: one in the equilibrium where high types sabotage low types and one
where they do not.

####  B.7.1 Option L1: High types sabotage only other high types, but no low
types

Consider a high type of value $v_{h}$ not currently sabotaging anyone and
sincere voting by everyone else so that $v_{h}=b_{h}(n+l+h)$ and
$v_{l}=b_{l}(n+l+h)$, and $S=hv_{h}+lv_{l}$. In this equilibrium, $h$ high
types vote $-b_{h}$ on $h-1$ all others (high and low types) and low type
start sabotaging $h$ high types with $-b_{h}$. This turns out to be the
relevant bound (see proof below).

$\displaystyle\frac{v_{l}}{S-hb_{h}(h-1)-b_{h}h}-\frac{v_{l}}{S-hb_{h}(h-1)-b_{h}(h-1)}$
(12) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{l}b_{h}}{(S-hb_{h}(h-1)-b_{h}h)(S-hb_{h}(h-1)-b_{h}(h-1))}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{l}b_{h}}{(S-hb_{h}h)(S-b_{h}h^{2}+b_{h})}$

####  B.7.2 Option L2: High types sabotage all other high types and low types

Consider a high type of value $v_{h}$ not currently sabotaging anyone and
sincere voting by everyone else so that $v_{h}=b_{h}(n+l+h)$ and
$v_{l}=b_{l}(n+l+h)$, and $S=hv_{h}+lv_{l}$. In this equilibrium, $h$ high
types vote $-b_{h}$ on $h-1$ other high types and vote $-b_{l}$ on $l$ low
types and low types start sabotaging $h$ high types with $-b_{h}$.

$\displaystyle\frac{v_{l}}{S-hb_{h}(h-1)-hb_{l}l-b_{h}h}-\frac{v_{l}}{S-hb_{h}(h-1)-hb_{l}l-b_{h}(h-1)}$
(13) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{l}b_{h}}{(S-hb_{h}(h-1)-hb_{l}l-b_{h}h)(S-hb_{h}(h-1)-hb_{l}l-b_{h}(h-1))}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{v_{l}b_{h}}{(S-hb_{h}h-hb_{l}l)(S+b_{h}-b_{h}h^{2}-hb_{l}l)}$

##### Proof: Order of Bounds.

To establish the correct order of bounds, we show that maximum marginal gain
for low types to sabotage high types is higher than the maximum marginal gain
for high types to sabotage low types.

$\displaystyle\frac{v_{h}b_{l}}{(S-hb_{h}(h-1)-b_{l}l)(S-hb_{h}(h-1)-b_{l}(l-1))}<\frac{v_{l}b_{h}}{(S-hb_{h}h)(S-b_{h}h^{2}+b_{h})}$
(14)

This inequality holds given $v_{l}<v_{h}$ and $b_{l}<b_{h}$ which are true by
construction. $\blacksquare$

This shows that low types have more to gain from sabotaging high types than
high types have to gain from sabotaging low types. Consequently, as the cost
for sabotage decrease, low types will switch from not sabotaging anyone to
sabotaging high types before high types will switch from sabotaging only other
high types to sabotaging all other high types and low types. As a consequence,
the relevant bound for low types switching to sabotaging high types is the one
where only high types sabotage high types but no low types (option H2 above).
Furthermore, the correct bound for high types to sabotage low types is one
where everyone already sabotages high types (option L1 above). Given that we
have now established a sequence of bounds, we establish that the set of
possible Nash equilibria is NE = {no sab, h sab h, l sab h, h sab l, l sab l}

###  B.8 Nash Equilibria

This contest has many Nash equilibria. Which equilibrium materializes depends
on the size of the groups $n$, $l$, and $h$, the costs $c_{s}$ and $c_{p}$,
and the performance gap $g$ between high and low types.

###### Proposition 1

In contests with a positive performance gap $g\geq 1$ between high and low
types, the following seven states are the set of possible Nash equilibria.

  1. (NE1)

$s_{i}=(0,0,0)$ for $i\in H$, and $s_{k}=(0,0,0)$ for $k\in L$. No one self-
promotes, and no one sabotages anyone.

  2. (NE2)

$s_{i}=(0,0,0)$ for $i\in H$, and $s_{k}=(0,0,1)$ for $k\in L$. Low types
engage in self-promotion, and no one sabotages anyone.

  3. (NE3)

$s_{i}=(0,0,1)$ for $i\in H$, and $s_{k}=(0,0,1)$ for $k\in L$. All agents
engage in self-promotion, and no one sabotages anyone.

  4. (NE4)

$s_{i}=(h-1,0,1)$ for $i\in H$, and $s_{k}=(0,0,1)$ for $k\in L$. All agents
engage in self-promotion, high types sabotage each other but do not sabotage
low types, and low types do not sabotage.

  5. (NE5)

$s_{i}=(h-1,0,1)$ for $i\in H$, and $s_{k}=(h,0,1)$ for $k\in L$. All agents
engage in self-promotion, high types sabotage each other but do not sabotage
low types, and low types sabotage only high types.

  6. (NE6)

$s_{i}=(h-1,l,1)$ for $i\in H$, and $s_{k}=(h,0,1)$ for $k\in L$. All agents
engage in self-promotion, high types sabotage all other agents of both types,
and low types sabotage only high types.

  7. (NE7)

$s_{i}=(h-1,l,1)$ for $i\in H$, and $s_{k}=(h,l-1,1)$ for $k\in L$. All agents
engage in self-promotion, and all agents sabotage all other agents of both
types.

##### Proof.

This follows directly from Lemma 1 (low types will engage in self-promotion
before high types do), Lemmas 4-6 which give an order of the bounds, and the
additional proof that the benefit for low types sabotaging high types is
larger than the benefit of high types sabotaging low types. $\blacksquare$

###  B.9 Revisiting the Performance Gap $g$

It is now interesting to revisit the condition of $g\geq 1$ for self-
promotion. It is clear that the closer $b_{h}$ is to $1$, the less high types
have to gain from self-promotion. Consequently, they will only engage in self-
promotion if the costs $c_{s}$ for doing so are increasingly smaller. In the
equilibrium condition NE4, for example, in which high types sabotage each
other and low types do not sabotage anyone, then $g\geq 1$ is violated if
$(n+h+l)b_{l}>(n+l+1)b_{h}$ and the resulting gain from self-promotion would
be lower for low types than high types. This would then lead to the
interesting case where low types do not self-promote while high types do.
Considering both self-promotion and sabotage, it is interesting to note the
following opposing effect of the performance gap: The larger the performance
gap, the more attractive self-promotion becomes for low types and the more
attractive sabotage becomes for high types. As a result, in contests with a
large performance gap, strategic behavior levels the playing field, thus
making the contest more attractive to low types.

From the argument above, we can draw another observation: Self-promotion acts
as an equalizing mechanism that reduces the performance difference between
high and low-skill agents and thus reduces incentives for high-skill agents to
sabotage (this can be seen by calculating and comparing Equation 8 in our
model with and without self-promotion).

###  B.10 Example and Additional Predictions

Let the prize be $M=\$5,000$ and let $h=10$, $l=30$, and $n=100$. Further, let
$b_{h}=0.8$ and $b_{l}=0.2$ (this corresponds to a common case of a finite
rating scale from 0-stars to 5-stars where low type agents produce contest
entries of 1-star quality and high type agents produce contest entries of
4-star quality). The resulting performance gap between low and high types is
then
$g=\frac{b_{h}}{b_{l}}\frac{n+1}{n+l+h}=\frac{0.8}{0.2}\frac{100+1}{100+30+10}>1$
which means that high types always have higher values than low types, even
when they are sabotaged. Figure A.I shows agent utility ($Mp$) as a function
of cost of sabotage $c_{s}$. The figure illustrates the transitions between
equilibria and their relative sizes in terms of the range of $c_{s}$ values
spanned. The figure suggests that there is one equilibrium in particular that
has a large basin of attraction that spans a wide range of $c_{s}$ values that
seem plausible in an online contest ($0.06 to $0.23): all agents self-promote
and high types sabotage other high types while low types do not sabotage
anyone (NE4).

Figure A.I: Equilibria and Utilities.

Note:  According to Lemma 4, all equilibria with some form of sabotage also
include self-promotion from all agents.

##  Appendix C Additional Empirical Analyses and Robustness Tests

###  C.1 Distribution of Skill and Contest Size

(a) Distribution of Skill

(b) Distribution of Contest Size

Figure A.II: Distributions of Skill and Contest Size.

Note: (a) We compute skill as the average rating of all previous submissions
by a designer. The skill distribution is right skewed (skewness: 0.21). (b)
Distribution of contest sizes (i.e., number of contest entries; mean: 302).

###  C.2 Career Length and Participation

On average, competitors have a career length from their first participation to
their last participation that spans 42 contests. During their active tenure,
they participate in some form in 9.1 contests, submit their own design in 1.5
contests and rate in 9.0. Submitting to a contest tends to be spaced out
between periods of just rating. On average, competitors have 1.4 “flips” in
which they go from rating as outsiders to rating as competitors, or from
rating as competitors to rating as outsiders. These changes are the basis of
our identification: we contrast rating behavior from weeks in which a
participant rates (9 contests on average) and weeks in which participants
submit their own design while also rating (on average 1.4 contests).

###  C.3 Rating Distribution of Contest Winners

We show the percentile scores of contest winners in Figure A.III.

Figure A.III: Percentile score of contest winners.

Note: Contest winners scored in the highest percentiles among all submissions
in a given week. The median percentile score was 98th percentile.

###  C.4 Heterogeneous Contestant Skill

##### Subsample.

As we move from our overall analysis (Table 3) to our examination of
heterogeneity by skill (Table 4), we have to restrict our analysis to a
subsample of the data: observations for which skill measures for both the
source (i.e., competitor casting the rating) and the target (i.e., competitor
who submitted the contest entry being rated) are available. This means we have
to exclude ratings by competitors who have never before submitted (and hence
source skill is unknown) and ratings of submissions by first-time submitters
(for whom target skill is unknown). In the interpretation of the main results,
one may consequently worry about sampling of subsets of the data. Table A.I
shows the main results across subsamples. We repeat the analysis of the full
data (Model 1), and the subset for which both skill measures are missing
(Model 4; i.e., the data used in Table 4).

Subsamples do not appear to drive our effects. The coefficients for sabotage
remain stable across the subsamples ranging from $-0.007$ to $-0.011$. Never-
before submitters are harsh raters and excluding these data increases the
positivity bias (Model 2). Competitors greet first-timers with more positive
ratings and the positivity bias decreases when we exclude those data (Model
3).

| Linear Probability
---|---
Dependent Variable: |  Sabotage: 0-Star Rating
|  | Exclude ratings by | Exclude ratings of |
| All data | never before submitters | submissions by first-timers | Exclude both
| (1) | (2) | (3) | (4)
Submitted to same contest: Yes | $-0.008^{***}$ | $-0.011^{***}$ | $-0.007^{***}$ | $-0.011^{***}$
| $(0.000)$ | $(0.000)$ | $(0.000)$ | $(0.000)$
Rate own submission: Yes | $-0.195^{***}$ | $-0.195^{***}$ | $-0.186^{***}$ | $-0.186^{***}$
| $(0.001)$ | $(0.001)$ | $(0.001)$ | $(0.001)$
Individual | Fixed | Fixed | Fixed | Fixed
Submission | Fixed | Fixed | Fixed | Fixed
Adj. R2 | $0.372$ | $0.391$ | $0.367$ | $0.386$
Num. obs. | 38,102,880 | 27,188,751 | 26,317,775 | 18,787,584
${}^{***}p<0.001$, ${}^{**}p<0.01$, ${}^{*}p<0.05$
Table A.I: Analysis of subsamples.

###  C.5 Robustness of Cluster Level of Standard Errors

In the main text, we cluster standard errors to account for possible
dependence of ratings of the same submissions and in the same contest. One may
alternatively be concerned about capturing autocorrelation at the rater level.
In table A.II we examine the robustness of our specification and cluster on
the individual level instead. While we do see some changes in the standard
errors, $p$-values are generally much smaller than $0.001$ (i.e., $1e-7$) so
that our substantial conclusions are not affected by the level of clustering
of standard errors.

| Linear Probability
---|---
Dependent Variable: | Sabotage | Self-Promotion
| 0-Star Rating | 5-Star Rating
| (1) | (2)
Submitted to same contest: Yes | $-0.057^{***}$ | $-0.000$
| $(0.008)$ | $(0.001)$
Rate own submission: Yes | $-0.179^{***}$ | $0.996^{***}$
| $(0.003)$ | $(0.016)$
Target skill | $-0.112^{***}$ | $0.008$
| $(0.011)$ | $(0.012)$
Source skill | $-0.049^{**}$ | $-0.105^{***}$
| $(0.017)$ | $(0.015)$
Target skill $\times$ Source skill | $0.007$ | $0.042^{***}$
| $(0.006)$ | $(0.006)$
Submitted to same contest: Yes |  |
$\times$ Target skill | $0.015^{***}$ |
| $(0.003)$ |
$\times$ Source skill | $0.010^{***}$ |
| $(0.003)$ |
Rate own submission: Yes $\times$ Source skill |  | $-0.080^{***}$
|  | $(0.009)$
Individual | Fixed | Fixed
Contest | Fixed | Fixed
Adj. R2 | $0.360$ | $0.208$
Num. obs. | 18,787,584
${}^{***}p<0.001$, ${}^{**}p<0.01$, ${}^{*}p<0.05$

Table A.II: Robustness test of heterogeneous effects by skill level with
errors clustered on the individual-level.

Note: Skill is measured as the average quality of all past submissions.
Contest-level fixed effects are used instead of submission-level as the
ability of agents is time invariant at the submission level. Standard errors
are in parentheses, clustered at the individual level.

###  C.6 Skill Measure

The analyses presented in the main body measure the idea generation skill of
individuals as the average quality of all past submissions. We repeat that
analysis using the quality of the best design they submitted in the past
(Table A.III and Figure A.IV). We find very similar results.

| Linear Probability
---|---
Dependent Variable: | 0-Star Rating | 5-Star Rating
| (1) | (2)
Submitted to same contest: Yes | $-0.055^{***}$ | $-0.000$
| $(0.004)$ | $(0.001)$
Rate own submission: Yes | $-0.179^{***}$ | $0.986^{***}$
| $(0.004)$ | $(0.007)$
Target max skill | $-0.066^{***}$ | $-0.001$
| $(0.002)$ | $(0.001)$
Source max skill | $-0.019^{***}$ | $-0.084^{***}$
| $(0.004)$ | $(0.003)$
Target max skill $\times$ Source max skill | $0.002$ | $0.027^{***}$
| $(0.001)$ | $(0.001)$
Submitted to same contest: Yes $\times$ Target max skill | $0.012^{***}$ |
| $(0.001)$ |
Submitted to same contest: Yes $\times$ Source max skill | $0.008^{***}$ |
| $(0.001)$ |
Rate own submission: Yes $\times$ Source ability |  | $-0.060^{***}$
|  | $(0.003)$
Individual | Fixed | Fixed
Contest | Fixed | Fixed
Adj. R2 | $0.360$ | $0.211$
Num. obs. | 18,787,584
${}^{***}p<0.001$, ${}^{**}p<0.01$, ${}^{*}p<0.05$

Table A.III: Estimates of heterogeneous effects using quality of the best
prior design as skill measure.

Note: Contest-level fixed effects are used instead of submission-level because
the skill of agents is time invariant at the submission level. Standard errors
are in parentheses, clustered at the contest level.

(a) Sabotage

(b) Self-Promotion

Figure A.IV: Strategic behavior by competitors of heterogeneous skill (max
skill).

Note: Panel A (Sabotage). Relative change in probability of rating 0-star when
competing compared to not competing across skill levels. Outer margins show
the distribution of data. There are over 15,000 (10,000) observations for
source (target) skill greater than 4.5. Panel B (Self-promotion). Relative
change in probability of rating 5-star when rating own submission compared to
submissions by others of the same skill (error band is 95% confidence
interval).

###  C.7 Sources of Sabotage

In this section we explore tenure, experience, and being a past winner as
different mechanism behind the decision to sabotage others (Table A.IV).
Similar to the analysis in the main text, we introduce interaction terms
between the Submitted to same contest dummy with different time-varying
measures of individual attributes. All models control for source and target
skill and thus need to be interpreted as drivers above and beyond the finding
that sabotage is predominantly used among high-skill competitors targeting
other high-skill competitors.

| Linear Probability
---|---
Dependent Variable: |  Sabotage: 0-Star Rating
| Tenure | Experience | Past Success
| (1) | (2) | (3)
Submitted to same contest: Yes | $-0.068^{***}$ | $-0.049^{***}$ | $-0.058^{***}$
| $(0.001)$ | $(0.001)$ | $(0.001)$
Rate on own submission: Yes | $-0.192^{***}$ | $-0.196^{***}$ | $-0.194^{***}$
| $(0.001)$ | $(0.001)$ | $(0.001)$
Tenure (in weeks) | $0.017^{***}$ |  |
| $(0.000)$ |  |
Rating experience (log) |  | $0.031^{***}$ |
|  | $(0.000)$ |
Submission experience (log) |  | $-0.010^{***}$ |
|  | $(0.000)$ |
Source is past winner |  |  | $-0.026^{***}$
|  |  | $(0.000)$
Target is past winner |  |  | $0.012^{***}$
|  |  | $(0.000)$
Submitted to same contest: Yes |  |  |
$\times$ Tenure | $0.000$ |  |
| $(0.000)$ |  |
$\times$ Rating experience |  | $-0.010^{***}$ |
|  | $(0.000)$ |
$\times$ Submission experience |  | $0.011^{***}$ |
|  | $(0.000)$ |
$\times$ Source is past winner |  |  | $0.021^{***}$
|  |  | $(0.004)$
$\times$ Target is past winner |  |  | $0.013^{***}$
|  |  | $(0.000)$
Skill controls | Yes | Yes | Yes
Individual | Fixed | Fixed | Fixed
Submission | Fixed | Fixed | Fixed
Adj. R2 | $0.391$ | $0.391$ | $0.391$
Num. obs. | 27,188,751
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$
Table A.IV: Regression of different mechanisms underlying strategic behavior.

First, we explore tenure on the platform (measured in months between the date
they signed up and the date a rating was submitted). We find no significant
effect (Model 1). We also tested the interaction with a quadratic term (not
shown) but find no effect either. Second, we find mixed results with regard to
submission and rating experience (Model 2). While community members with
higher rating experience appear to be less likely to sabotage, those with
higher submission experience appear to be more likely. This is consistent with
the interpretation that community members with a strong competitive motivation
(making many submissions) are more likely to act strategically compared to
community members who may be more socially motivated (evaluating many
submissions by others). Third, we find past winners are significantly more
likely to sabotage and are more likely to be the targets of sabotage.

##  Appendix D Leniency as Variation within Contests

Table A.V shows regression on the contest-level using standard deviation of
ratings a user casts as dependent variable. We find that when individuals have
submitted to the same contest, they make better use of the full rating
spectrum and submit ratings with a higher standard deviation (Model 1:
$\beta=0.037;p<0.001$). The effect is amplified by skill so that higher-
skilled individuals submit ratings with an even higher standard deviation
(positive interaction between skill and having submitted to the same contest
in Model 2: $\beta=0.017;p<0.001$).

| OLS
---|---
Dependent Variable: | Std.Dev. of Ratings
| (1) | (2)
Submitted to same contest: Yes | $0.037^{***}$ | $0.004$
| $(0.002)$ | $(0.009)$
Source ability |  | $-0.031^{***}$
|  | $(0.007)$
Submitted to same contest: Yes |  | $0.017^{***}$
$\times$ Source ability |  | $(0.005)$
Individual | Fixed | Fixed
Contest | Fixed | Fixed
Adj. R2 | $0.299$ | $0.299$
Num. obs. | $439,882$ | $439,882$
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$
Table A.V: User-contest level analysis of standard deviations of ratings cast.

###  D.1 Substitution of sabotage and self-promotion

| Linear Probability
---|---
Dependent Variable: |  Sabotage: 0-Star Rating
| (1) | (2)
Submitted to same contest: Yes | $-0.006^{***}$ | $-0.057^{***}$
| $(0.000)$ | $(0.000)$
Rate own submission: Yes | $-0.194^{***}$ | $-0.205^{***}$
| $(0.001)$ | $(0.000)$
Self-promoted in this contest | $-0.002^{**}$ | $0.031^{***}$
| $(0.001)$ | $(0.001)$
Individual | Fixed | Fixed
Submission | Fixed | Fixed
Adj. R2 | $0.372$ | $0.064$
Num. obs. | 38,102,880
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$
Table A.VI: Regression models investigating whether individuals use sabotage
and self-promotion together or substitute one for the other.

###  D.2 Robustness: Dyad Fixed Effects

As a robustness test, we investigate dyadic patterns underlying our findings
of sabotage. Research on dyadic rivalry may be an alternative explanation
(Kilduff et al., 2016; Grad et al., 2023). For example, individuals may
“sabotage” others with whom they have competed in the past and against whom
they have lost. One may hence wonder if there are (or emerge over time) dyads
in which individuals systematically sabotage others. Some dyads may rate each
other with 0-stars whether or not they are competitors in the current contest
and rivalry could be an alternative explanation to the strategic motivation
(i.e., the behavior is founded on the subjective feeling of rivalry instead of
the strategic motivation to increase ones chance of winning the contest
prize). If so, that may lead to an overestimation of our effects in the sense
that we attribute it to being “strategic” rather than “rivalrous”. To
investigate this alternative explanation, we estimate a variation of our main
model with dyad-level fixed effects (Table A.VII). Since rivalry is generally
assumed to require some time to form (Kilduff et al., 2016; Grad et al.,
2023), we restrict our analysis to dyads with ten or more encounters (e.g.,
$A$ rating five of $B$’s submissions and $B$ rating five of $A$’s
submissions). Note that the effect we report in the table is driven by many
fewer observations: only those dyads who rate each other at least once in the
same contest and once outside the same contest contribute to the parameter
estimate. Controlling for unobserved dyadic rivalries through these dyad-level
fixed effects (instead of our usual individual- and submission-level fixed
effects) we find that likelihood to sabotage others when competing (the main
effect) increases by more than half ($-0.070$ vs. $-0.043$). If the “sabotage”
effect we report in our estimates were driven by unobserved rivalries, then we
would expect the effect of rating ones competitors to decrease. Furthermore,
we find significant positive effects for sabotage committed by high-skilled
targeting other high-skilled just as in the analysis reported in the main text
(interaction term coefficients $\beta=0.015;p<0.001$ for sources and
$\beta=0.002;p=0.053$ for targets). This suggests that sabotage happens
outside, and on top of, any pre-existing dyadic rivalries. That is, even
within the same dyad, we still find a difference in the rating behavior
between competing for the same prize vs. not competing. In summary, dyadic
rivalry patterns are not sufficient to explain the strategically motivated
sabotage.

| Linear Probability
---|---
Dependent Variable: |  Sabotage: 0-Star Rating
| 10+ Encounters
| (1)
Submitted to same contest: Yes | $-0.043^{***}$
| $(0.004)$
Target skill | $-0.029^{***}$
| $(0.008)$
Source skill | $-0.025^{**}$
| $(0.009)$
Target skill $\times$ Source skill | $-0.006^{\cdot}$
| $(0.004)$
Submitted to same contest: Yes |
$\times$ Target skill | $0.002^{\dagger}$
| $(0.001)$
$\times$ Source skill | $0.015^{***}$
| $(0.001)$
Source-Target Dyad | Fixed
Adj. R2 | $0.448$
Num. obs. | 36,94,941
Num. dyads | 22,9404
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$; ${}^{\cdot}p<0.1$
Table A.VII: Robustness tests using dyad fixed effects.

###  D.3 Natural Experiment 1: Prize Increase

Figure A.V: Screenshot natural experiment sabotage.

Note: Screenshot of the Threadless website showing the announcement of the
prize increase posted on July 12, 2005.

###  D.4 Natural Experiment 2: Scoring Rule Change

Figure A.VI: Screenshot natural experiment self-promotion

Note: Screenshot of the Threadless website showing the announcement of the
scoring rule change posted on April 29, 2005.

###  D.5 Long-Term Participation

Table 6 in the main text shows coefficient estimates for the hazard rate of
participation in the next round. The hazard rate can also be shown visually
(Figure A.VII). The figure contrasts how high- versus low-skilled competitors
(10th vs. 90th percentile, respectively) react to receiving high versus low
levels of sabotage (also 10th vs. 90th percentile, respectively).

Figure A.VII: Likelihood of participating in a future contest.

Note:  Likelihood for high skilled (90th percentile) and low skilled (10th
percentile) individuals based on experience in the current contest for
individuals who received high (90th percentile) and low (10th percentile)
levels of sabotage (cumulative hazard; based on Model 1 from Table 6; controls
include rating received, number of ratings received, amount of competition,
quality of competition)..

##  Appendix E Effect of Strategic Rating on Selection of Contest Winners

Does strategic rating behavior affect the rank-order of contest submissions,
especially given the generally large contests we observe? To assess the impact
of self-promotion, we exclude all ratings from designers who rated their own
designs, recalculate average ratings for submissions, and then re-rank all
submissions based on the new averages. We find that, keeping everything else
constant, in seven out of 511 contests (1.4%), the winner of the contest
changes; and in 28 of 511 contests (5.5%), there is a change in at least one
of the top three ranks. To analyze the effect of sabotage, we exclude all
ratings from competitors as they might be strategically motivated. Note that
this also includes votes from designers on their own designs, i.e., it
contains the previous reported effects. We find that in 12% of the contests,
the winner of the contest changes; and that in 48% of the contests, there is a
change in the top three ranks. The effect is especially pronounced in close
contests where the contest winner would change in 25% of cases and 65% would
see a change in the top three. This can be considered an upper boundary for
the effect of strategic behavior via the rating mechanism in our setting.

This change in rankings, however, may be simply due to the fact that removing
any ratings changes the resulting ranking. So to complement this first test,
we compare the effect of removing the ratings of potentially strategic raters,
to the effect of removing the ratings of an equal amount of randomly selected
raters. Removing random raters, contest winner changes on average 10.7% of
cases across our bootstrap simulation, while contest winners change 12% of the
time when removing ratings by potentially strategic raters. In close contests,
contest winner changes 13.8% in the null model vs. 25% winner change when
removing potentially strategic raters. Removing strategic ratings affect
outcomes more than just removing random raters which lends further support to
our claim of the important role of strategic rating. Overall, these two
analyses suggest that strategic behavior of idea generators during idea
selection, does affect Threadless’ ability to identify the most promising
ideas. The effect is strongest in close competition where up to 25% of contest
winners may change and around two-thirds of contests would have changes in the
top-three.

To further quantify the effect of strategic rating, we compute a baseline of
how much a contests rank-order would change if the ratings of some randomly
chosen individuals were removed. For each contest, we compute the number of
competitors who did cast potentially strategic ratings, and select the same
amount of raters at random. To remove the same number of ratings, we need to
remove about 20% more raters since competitors who submitted to the same
contest rate more submissions that those who did not. We remove their ratings
and recompute the ranking to see how much has changed. We repeat this 500
times for all contests in our data in order to get a robust null model of how
much rankings would change after removing random raters.


# Leptoquark explanation of $h\to\mu\tau$ and muon $(g-2)$

Seungwon Baek<EMAIL_ADDRESS>School of Physics, KIAS, 85 Hoegiro, Seoul
02455, Korea  Kenji Nishiwaki<EMAIL_ADDRESS>School of Physics, KIAS, 85
Hoegiro, Seoul 02455, Korea

###### Abstract

We consider lepton flavor violating Higgs decay, specifically $h\to\mu\tau$,
in a leptoquark model. We introduce two scalar leptoquarks with the
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ quantum numbers, $(3,2,7/6)$ and
$(3,2,1/6)$, which do not generate dimension-4 operators mediating proton
decay. They can mix with each other by interactions with the standard model
Higgs. The constraint from the charged lepton flavor violating process,
$\tau^{-}\to\mu^{-}\gamma$, is very strong when only one leptoquark
contribution is considered. However, we demonstrate that significant
cancellation is possible between the two leptoquark contributions. We show
that we can explain the CMS (ATLAS) excess in $h\to\mu\tau$. We also show that
muon $(g-2)$ anomaly can also be accommodated.

###### pacs:

12.60.Fr, 12.60.-i, 13.85.Qk

††preprint: KIAS-P15052

##  I Introduction

Leptoquarks (LQs) are scalar particles which carry both baryon and lepton
numbers Buchmuller:1986zs . They appear in gauge theories with “unified” gauge
groups, such as Pati-Salam model, $SU(5)$ grand unification, etc.

Since LQs are strongly interacting particles which can decay semileptonically,
their masses are strongly bounded by the LHC experiments, such as ATLAS and
CMS. For the third-generation scalar LQs, the ATLAS group excludes the mass in
the range $m_{\text{LQ}}<625\,\text{GeV}$ and
$200\,\text{GeV}<m_{\text{LQ}}<640\,\text{GeV}$ at $95\%$ confidence level
(C.L.) based on their $8\,\text{TeV}$ data, assuming $100\%$ branching
fractions into $b\nu_{\tau}$ and $t\nu_{\tau}$, respectively Aad:2015caa . On
the other hand, the CMS group had reported various $8\,\text{TeV}$ bounds at
$95\%$ C.L. on $m_{\text{LQ}}$ as $m_{\text{LQ}}\,{>}\,740\,\text{GeV}$,
$m_{\text{LQ}}\,{>}\,650\,\text{GeV}$ and
$m_{\text{LQ}}\,{>}\,685\,\text{GeV}$ with assumptions of $100\%$ branching
fractions into $b\tau$, $t\nu_{\tau}$ and $t\tau$, respectively
Khachatryan:2014ura ; Khachatryan:2015bsa .

We note that the CMS excess of $eejj$ and $e\nu jj$ CMSeejj  can also be
interpreted as a signal of the first generation LQ with mass about 650 GeV. An
example of detailed study of LQ models for the excess can be found in
Queiroz:2014pra .

In the standard model (SM), lepton flavor violating (LFV) Higgs decay channels
are absent at tree level and highly suppressed by small neutrino masses and
the GIM mechanism at loop level. Therefore, once they are observed with
sizable branching fractions, they indicate a clear signal of new physics
beyond the SM. The CMS collaboration reported the LFV Higgs decay branching
fraction, using the 19.7 fb-1 of $\sqrt{s}=8$ TeV,

$\displaystyle\mathcal{B}(h\to\mu\tau)={(0.84^{+0.39}_{-0.37})}\%,$ (I.1)

which deviates $2.4\sigma$ from zero Khachatryan:2015kon . Here, $\mu\tau$
means the inclusive final state consisting of $\mu^{+}\tau^{-}$ and
$\mu^{-}\tau^{+}$. Although recent ATLAS measurement, using the 20.3 fb-1 of
$\sqrt{s}=8$ TeV,

$\displaystyle\mathcal{B}(h\to\mu\tau)=(0.77\pm 0.62)\%,$ (I.2)

does not show a significant deviation from the SM Aad:2015gha , it is at least
consistent with the CMS result. If confirmed by the future data at LHC Run II
which can probe down to $\sim 10^{-3}$, it would be a clear signal requiring
new physics beyond the SM. There are several model-independent Harnik:2012pb ;
Blankenburg:2012ex ; Davidson:2012ds ; Kopp:2014rva ; Celis:2014roa ;
Lee:2014rba ; deLima:2015pqa ; Varzielas:2015iva ; Bhattacherjee:2015sia ;
Mao:2015hwa ; He:2015rqa ; Goto:2015iha ; Altmannshofer:2015esa  and also
model-dependent studies Pilaftsis:1992st ; Arhrib:2012mg ; Falkowski:2013jya ;
Dery:2014kxa ; Campos:2014zaa ; Sierra:2014nqa ; Heeck:2014qea ;
Crivellin:2015mga ; Dorsner:2015mja ; Omura:2015nja ; Crivellin:2015lwa ;
Das:2015zwa ; Chiang:2015cba ; Crivellin:2015hha ; Cheung:2015yga ;
Botella:2015hoa ; Baek:2015fma  to accommodate this deviation.

The theoretical and experimental sensitivity of the anomalous magnetic moment
of the muon, i.e. $(g-2)_{\mu}$, has reached to probe the electroweak scale.
State of the art calculations in the SM cannot explain the experimental
result, and there is about 3$\sigma$ discrepancies between them
Jegerlehner:2009ry ; Benayoun:2012wc :

$\displaystyle\Delta a_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=(299\pm
90~{}~{}\text{to}~{}~{}394\pm 84)\times 10^{-11},$ (I.3)

which also calls for new physics models.

In this paper we consider a LQ model as an explanation of the LFV Higgs decay,
$h\to\mu\tau$ and muon $(g-2)$ anomaly. Considering the proton decay
constraints, only two types of $SU(2)_{L}$ doublet leptoquarks are favored. We
assume both of them are realized in nature. We first show that a strong
constraint from $\tau^{-}\to\mu^{-}\gamma$ can be alleviated significantly due
to cancellations between the top and bottom quark contributions.111Note that
similar discussions in the context of LQs are found in Dorsner:2015mja ;
Cheung:2015yga . We show that there is allowed parameter space to accommodate
the $h\to\mu\tau$ anomaly and $(g-2)_{\mu}$. A smoking gun signal which
distinguishes our model from other models would be the direct LQ production at
colliders. A promising signature at the LHC is the pair production of LQs
decaying into a quark and a lepton, where the decay pattern is so
characteristic. Especially, components with $+2/3$ electric charge, named as
$Y_{1}$ and $Y_{2}$ later, should be relatively light and couple to the bottom
quark, the electron and the muon for the explanation of the excess in
$h\to\mu\tau$ and the muon $(g-2)$ with circumventing the bound from
$\tau^{-}\to\mu^{-}\gamma$. Therefore, the smoking gun final states in our
model are $b\overline{b}\,\tau^{-}\tau^{+}$ and
$b\overline{b}\,\mu^{-}\mu^{+}$.

The paper is organized as follows. In Sec. II, we introduce our model. In Sec.
III, we consider $\tau^{-}\to\mu^{-}\gamma$ constraint. In Sec. IV, we
consider $h\to\mu\tau$ signal. In Sec. V, we show that we show that we can
accommodate $(g-2)_{\mu}$. In Sec. VI, we summarize and conclude.

##  II The Model

Among the possible LQs which have renormalizable interactions with the SM
fermions, only $R_{2}$ and $\widetilde{R}_{2}$ in the notation of
Buchmuller:1986zs  do not have problem with the constraint from the proton
decay within renormalizable perturbation theory Arnold:2013cva ;
Queiroz:2014pra . They are in the representation

$\displaystyle R_{2}(3,2,7/6),\quad\widetilde{R}_{2}(3,2,1/6),$ (II.1)

in the SM gauge group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$.222In the
case of non-$SU(2)_{L}$-doublet LQs, we can write down gauge-invariant
dimension-four operators generating rapid proton decay. The $SU(2)_{L}$
doublet ones do not allow such dangerous operators at renormalizable level.
However, as discussed in Arnold:2013cva ; Queiroz:2014pra , constraints from
dimension-five effective operators (generating proton decay) are still severe,
where $m_{\text{LQ}}$ should be greater than around $10^{5}\,\text{TeV}$ even
when the cutoff scale is equal to the Planck scale. A remedy for reducing
$m_{\text{LQ}}$ is to introduce a new symmetry prohibiting the operators. In
this paper, we do not consider constraints from the proton decay caused by
higher dimensional operators.

Assuming both of them exist in nature at renormalizable level, they interact
with quarks and leptons via the interaction Lagrangian

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\lambda_{u}^{ij}\overline{u}_{R}^{i}R_{2}^{T}\epsilon\,L_{L}^{j}-\lambda_{e}^{ij}\overline{e}_{R}^{i}R_{2}^{\dagger}Q_{L}^{j}-\lambda_{d}^{ij}\overline{d}_{R}^{i}\widetilde{R}_{2}^{T}\epsilon\,L_{L}^{j}+{h.c.},$
(II.2)

where we have suppressed color indices and $\epsilon\,(\equiv i\sigma^{2})$ is
the two-by-two antisymmetric matrix with $\epsilon^{12}=1$. The scalar
potential is given by

$\displaystyle V$ $\displaystyle=$
$\displaystyle\mu_{H}^{2}|H|^{2}+\mu_{2}^{2}|R_{2}|^{2}+\widetilde{\mu}_{2}^{2}|\widetilde{R}_{2}|^{2}$
(II.3) $\displaystyle+$
$\displaystyle\lambda_{H}|H|^{4}+\lambda_{2}|R_{2}|^{4}+\widetilde{\lambda}_{2}|\widetilde{R}_{2}|^{4}+\lambda_{HR}|H|^{2}|R_{2}|^{2}+\widetilde{\lambda}_{HR}|H|^{2}|\widetilde{R}_{2}|^{2}$
$\displaystyle+$
$\displaystyle\lambda_{H2}R_{2}^{\dagger}HH^{\dagger}R_{2}+\widetilde{\lambda}_{H2}\widetilde{R}_{2}^{\dagger}HH^{\dagger}{\widetilde{R}_{2}}+\left(\lambda_{\rm
mix}R_{2}^{\dagger}H\widetilde{R}_{2}\epsilon\,H+h.c.\right),$

where $H(1,2,1/2)$ is the SM Higgs doublet. $R_{2}$ and $\widetilde{R}_{2}$
fields can be decomposed into $SU(2)_{L}$ components,

$\displaystyle R_{2}=\left(\begin{array}[]{c}V\\\
Y\end{array}\right),\quad\widetilde{R}_{2}=\left(\begin{array}[]{c}\widetilde{Y}\\\
\widetilde{Z}\end{array}\right).$ (II.8)

After the Higgs gets vacuum expectation value (vev), $v\,(\simeq
246\,\text{GeV})$, we can write

$\displaystyle H=\left(\begin{array}[]{c}0\\\
\frac{1}{\sqrt{2}}(v+h)\end{array}\right),$ (II.11)

in the unitary gauge. Then, the masses of $V$ and $\widetilde{Z}$ are given by

$\displaystyle m_{V}^{2}=\mu_{2}^{2}+{1\over 2}\lambda_{HR}v^{2},\quad
m_{\widetilde{Z}}^{2}=\widetilde{\mu}_{2}^{2}+\frac{1}{2}\widetilde{\lambda}_{HR}v^{2}+\frac{1}{2}\widetilde{\lambda}_{H2}v^{2}.$
(II.12)

The mass terms of $Y$ and $\widetilde{Y}$ are written as

$\displaystyle{\cal L}_{\rm
mass}(Y,\widetilde{Y})=-\left(\begin{array}[]{cc}Y^{\dagger}&\widetilde{Y}^{\dagger}\end{array}\right)\left(\begin{array}[]{cc}\mu_{2}^{2}+\frac{1}{2}\lambda_{HR}v^{2}+\frac{1}{2}\lambda_{H2}v^{2}&\frac{1}{2}\lambda_{\rm
mix}v^{2}\\\ \frac{1}{2}\lambda_{\rm
mix}v^{2}&\widetilde{\mu}_{2}^{2}+\frac{1}{2}\widetilde{\lambda}_{HR}v^{2}\end{array}\right)\left(\begin{array}[]{cc}Y\\\
\widetilde{Y}\end{array}\right).$ (II.18)

The mass eigenstates, $Y_{1},Y_{2}$ (with the electromagnetic charge $+2/3$)
are mixture of $Y$ and $\widetilde{Y}$ with mixing angle $\alpha_{Y}$,

$\displaystyle\left(\begin{array}[]{cc}Y\\\
\widetilde{Y}\end{array}\right)=\left(\begin{array}[]{cc}c_{Y}&s_{Y}\\\
-s_{Y}&c_{Y}\end{array}\right)\left(\begin{array}[]{cc}Y_{1}\\\
Y_{2}\end{array}\right)\equiv O\left(\begin{array}[]{cc}Y_{1}\\\
Y_{2}\end{array}\right),$ (II.27)

where $c_{Y}=\cos\alpha_{Y},s_{Y}=\sin\alpha_{Y}$.

As we will see, large $\alpha_{Y}$ and large mass splitting between $V$ and
$Y_{i}$ are favored to satisfy the experimental constraints and also to
enhance $h\to\mu\tau$. Concretely speaking, the relation $m_{Y_{i}}\sim
m_{V}/6$ will be imposed to avoid the bound from $\tau^{-}\to\mu^{-}\gamma$
naturally with sizable couplings, which are required for explanations of the
excess in $h\to\mu\tau$.

Here, we look into the mass matrix in Eq. (II.18) and discuss whether we can
realize the mass hierarchy as $m_{Y_{i}}\sim m_{V}/6$ with a large mixing in
$\alpha_{Y}$ in our setup. A key point is that the $(1,1)$ component of the
mass matrix is rephrased as $m_{V}^{2}+\frac{1}{2}\lambda_{H2}v^{2}$. Then,
when the following relations are realized,

$\displaystyle m_{V}^{2},\,\lambda_{H2}v^{2}\ >\
\lambda_{\text{mix}}v^{2},\,\widetilde{\mu}_{2}^{2}+\frac{1}{2}\widetilde{\lambda}_{HR}v^{2},$
(II.28)

and a cancellation occurs between $m_{V}^{2}$ and
$\frac{1}{2}\lambda_{H2}v^{2}$ with a negative $\lambda_{H2}$, the relation
$m_{Y_{i}}\sim m_{V}/6\,(i=1,2)$ can be realized. In addition, if the off-
diagonal terms are comparable with diagonal ones, a large mixing angle in
$\alpha_{Y}$ is expected. For example, if the (1,1) and (2,2) components are
of similar size with $\sim{\cal O}(1)$ TeV2 and $\lambda_{\rm mix}\sim{10}$,
we get ${m_{Y_{1,2}}\gtrsim}~{}0.7~{}{\rm TeV}$ and a maximal $\alpha_{Y}$,
which can obviously avoid the current direct search bound on third generation
LQs. However, when $m_{V}$ is multi TeV, a realization of such cancellation
between $m_{V}^{2}$ and $\frac{1}{2}\lambda_{H2}v^{2}$ would get to be
nontrivial within perturbative $\lambda_{H2}$. To further enhance mass
difference between $V$ and $Y_{i}$ and/or the mixing angle $\alpha_{Y}$, we
can implicitly assume additional contributions via higher dimensional
operators such as

$\displaystyle\frac{\phi}{\Lambda}R_{2}^{\dagger}HH^{\dagger}R_{2},\quad\frac{\phi}{\Lambda}R_{2}^{\dagger}H\widetilde{R}_{2}\epsilon\,H,$
(II.29)

where $\phi$ is a new singlet with a large vev as
$\langle\phi\rangle>\Lambda$.

Although there is no apparent symmetry which leads to the mass ratio
$m_{Y_{i}}\sim m_{V}/6$, the UV complete grand unified theory (GUT) or flavor
theory into which our low energy effective theory is embedded have larger
symmetry and we expect they guarantee the mass ratio without fine tuning. The
high energy theory will also generate the dimension-five operators in (II.29).

Finally, we comment on the case with a small mixing angle $\alpha_{Y}$, which
corresponds to the possibility that no sizable cancellation occurs between the
terms $m_{V}^{2}$ and $\frac{1}{2}\lambda_{H2}v^{2}$. Even in this case, (at
least) one mass eigenstate can be light as the relation $m_{Y_{i}}\sim
m_{V}/6$ being fulfilled. But, as we will see in Sec. III, we should accept a
larger hierarchy between two leptoquark couplings to circumvent the bound from
$\tau^{-}\to\mu^{-}\gamma$. To make matters worse, as discussed in Sec. IV,
such hierarchical couplings are inappropriate for explaining the excess in
$h\to\mu\tau$.

##  III $\tau^{-}\to\mu^{-}\gamma$

In this section, we consider the constraints from the charged lepton flavor
violating processes. Since we are interested in $2\leftrightarrow 3$
transitions, we restrict ourselves only to $\tau^{-}\to\mu^{-}\gamma$ decay.
Our study can be applied to other LFVs, such as $\mu^{-}\to e^{-}$ or
$\tau^{-}\to e^{-}$ transitions, similarly. However, we assume they are
sufficiently suppressed by small LFV couplings.

Figure 1: Feynman diagrams for $\tau^{-}\to\mu^{-}\gamma$. The photon line can
be attached to any charged particles, and there are four possibilities.

The effective Hamiltonian for $\tau^{-}\to\mu^{-}\gamma$ is written as

$\displaystyle{\cal H}_{\rm
eff}=C_{R}^{\gamma}~{}\overline{\mu}_{L}\sigma^{\mu\nu}F_{\mu\nu}\tau_{R}+C_{L}^{\gamma}~{}\overline{\mu}_{R}\sigma^{\mu\nu}F_{\mu\nu}\tau_{L},$
(III.1)

where $C_{R,L}^{\gamma}$ are Wilson coefficients and
$F_{\mu\nu}\,(=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})$ is the photon
field strength tensor.

The Feynman diagrams for $\tau^{-}\to\mu^{-}\gamma$ are shown in Fig. 1. We
note that in our model, the chirality flip appearing in (III.1) can occur
inside the loop. Therefore the amplitudes can be proportional to $m_{t}$ or
$m_{b}$ instead of small masses from the external lines, $m_{\tau}$ or
$m_{\mu}$. This is the main reason that this LFV process becomes a very strong
constraint in ordinary third generation LQ models.

The Wilson coefficients $C_{R,L}^{\gamma}$ can be calculated from the diagrams
in Fig. 1:

$\displaystyle C_{R}^{\gamma}$ $\displaystyle=$
$\displaystyle\frac{N_{c}e}{32\pi^{2}m_{V}^{2}}\Bigg{[}\Big{(}\lambda_{e}^{23}\lambda_{e}^{33*}m_{\mu}+\lambda_{u}^{32*}\lambda_{u}^{33}m_{\tau}\Big{)}\Big{(}\frac{2}{3}I_{1}(x)+\frac{5}{3}J_{1}(x)\Big{)}+\lambda_{u}^{32*}\lambda_{e}^{33*}m_{t}\Big{(}\frac{2}{3}I_{2}(x)+\frac{5}{3}J_{2}(x)\Big{)}\Bigg{]}$
$\displaystyle+$
$\displaystyle\sum_{j=1,2}\frac{N_{c}e}{32\pi^{2}m_{Y_{j}}^{2}}\Bigg{[}\Big{(}\lambda_{e}^{23}\lambda_{e}^{33*}O_{1j}^{2}m_{\mu}+\lambda_{d}^{32*}\lambda_{d}^{33}O_{2j}^{2}m_{\tau}\Big{)}\Big{(}-\frac{1}{3}I_{1}(y_{j})+\frac{2}{3}J_{1}(y_{j})\Big{)}$
$\displaystyle+$
$\displaystyle\lambda_{d}^{32*}\lambda_{e}^{33*}O_{1j}O_{2j}m_{b}\Big{(}-\frac{1}{3}I_{2}(y_{j})+\frac{2}{3}J_{2}(y_{j})\Big{)}\Bigg{]},$
$\displaystyle C_{L}^{\gamma}$ $\displaystyle=$
$\displaystyle\frac{N_{c}e}{32\pi^{2}m_{V}^{2}}\Bigg{[}\Big{(}\lambda_{u}^{32*}\lambda_{u}^{33}m_{\mu}+\lambda_{e}^{23}\lambda_{e}^{33*}m_{\tau}\Big{)}\Big{(}\frac{2}{3}I_{1}(x)+\frac{5}{3}J_{1}(x)\Big{)}+\lambda_{e}^{23}\lambda_{u}^{33}m_{t}\Big{(}\frac{2}{3}I_{2}(x)+\frac{5}{3}J_{2}(x)\Big{)}\Bigg{]}$
(III.2) $\displaystyle+$
$\displaystyle\sum_{j=1,2}\frac{N_{c}e}{32\pi^{2}m_{Y_{j}}^{2}}\Bigg{[}\Big{(}\lambda_{d}^{32*}\lambda_{d}^{33}O_{2j}^{2}m_{\mu}+\lambda_{e}^{23}\lambda_{e}^{33*}O_{1j}^{2}m_{\tau}\Big{)}\Big{(}-\frac{1}{3}I_{1}(y_{j})+\frac{2}{3}J_{1}(y_{j})\Big{)}$
$\displaystyle+$
$\displaystyle\lambda_{e}^{23}\lambda_{d}^{33}O_{1j}O_{2j}m_{b}\Big{(}-\frac{1}{3}I_{2}(y_{j})+\frac{2}{3}J_{2}(y_{j})\Big{)}\Bigg{]},$

where $N_{c}=3$ is the color factor, $x=m_{t}^{2}/m_{V}^{2}$, and
$y_{i}=m_{b}^{2}/m_{Y_{i}}^{2}$. The loop functions are obtained to be

$\displaystyle I_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{2+3x-6x^{2}+x^{3}+6x\log x}{12(1-x)^{4}},$ $\displaystyle
J_{1}(x)$ $\displaystyle=$ $\displaystyle\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\log
x}{12(1-x)^{4}},$ $\displaystyle I_{2}(x)$ $\displaystyle=$
$\displaystyle\frac{-3+4x-x^{2}-2\log x}{2(1-x)^{3}},$ $\displaystyle
J_{2}(x)$ $\displaystyle=$ $\displaystyle\frac{1-x^{2}+2x\log x}{2(1-x)^{3}}.$
(III.3)

The branching ratio of $\tau^{-}\to\mu^{-}\gamma$ is then

$\displaystyle{\cal B}(\tau^{-}\to\mu^{-}\gamma)$ $\displaystyle=$
$\displaystyle\frac{\tau_{\tau}(m_{\tau}^{2}-m_{\mu}^{2})^{3}}{4\pi
m_{\tau}^{3}}\left(|C_{R}^{\gamma}|^{2}+|C_{L}^{\gamma}|^{2}\right),$ (III.4)

where $\tau_{\tau}=87.03$ $\rm\mu m$ is the lifetime of $\tau$. The current
experimental bound is BaBar:tau2mug

$\displaystyle{\cal B}(\tau^{-}\to\mu^{-}\gamma)<4.4\times 10^{-8}.$ (III.5)

This corresponds to

$\displaystyle|C_{R}^{\gamma}|^{2}+|C_{L}^{\gamma}|^{2}<\left(4.75\times
10^{-10}\over{\rm GeV}\right)^{2}.$ (III.6)

Figure 2: Contour plots for $(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}$ which
is required for exact cancellations of $\tau^{-}\to\mu^{-}\gamma$ in
$(m_{Y_{2}},m_{Y_{1}})$-plane through Eq. (III.7). From top-left to bottom,
the sine of the mixing angle $\sin{\alpha_{Y}}$ is chosen as $1/\sqrt{2}$,
$0.3$ and $0.2$, respectively. $m_{V}$ is set as $m_{V}=6\,m_{Y_{2}}$ and
$m_{Y_{1}}$ is formulated as $m_{Y_{1}}=a\,m_{Y_{2}}$ by use of the factor
$a$, where the range $[1.1,6.0]$ is considered in the three plots.

For the discussion of $\tau^{-}\to\mu^{-}\gamma$, we assume $C_{L}^{\gamma}=0$
for simplicity. If we consider a single leptoquark contribution from $V$,
${\cal B}(\tau^{-}\to\mu^{-}\gamma)$ gives lower mass bound $0.85,3.7,14,42$
TeV, for $\lambda_{e}^{33}=0.001,0.01,0.1,1$, respectively, where we took
$\lambda_{u}^{32}=0.35$. With these parameters, we obtain too small
contribution to $h\to\mu\tau$ as was noticed in Dorsner:2015mja .

Since we introduce both $R_{2}$ and $\widetilde{R}_{2}$, we can have diagrams
with the chirality flip inside the $b$-quark loop, which generate terms
proportional to $m_{b}$. The $Y_{i}-b$ contributions are naively expected to
be smaller than $V-t$ contribution by factor $m_{b}/m_{t}\sim 1/35$. However,
since $C_{R,L}^{\gamma}$ are proportional to $m_{f}/m_{\rm LQ}^{2}$ ($f=t,b$)
as can be seen in (III.2), if $m_{Y_{i}}\sim m_{V}/6$, cancellations between
$t$ and $b$ contributions can occur naturally. Note that a nonzero mixing
between $Y$ and $\widetilde{Y}$ is mandatory for a natural cancellation since
in the limit $\sin{\alpha_{Y}}\to 0$ the contributions being proportional to
$m_{b}$ turn out to be zero. Neglecting small terms proportional to $m_{\tau}$
or $m_{\mu}$, an exact cancellation in $C_{R}^{\gamma}$ occurs when the
following condition is held,

$\displaystyle\frac{\lambda_{u}^{32*}}{\lambda_{d}^{32*}}={-\sum_{i=1}^{2}O_{1i}O_{2i}\,\frac{m_{b}}{m_{t}}\left(\frac{m_{V}}{m_{Y_{i}}}\right)^{2}\frac{-\frac{1}{3}I_{2}(y_{i})+\frac{2}{3}J_{2}(y_{i})}{+\frac{2}{3}I_{2}(x)+\frac{5}{3}J_{2}(x)}.}$
(III.7)

In Fig. 2, the values of the ratio
$(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}$ which are required for exact
cancellations of $\tau^{-}\to\mu^{-}\gamma$ are shown as
$(m_{Y_{2}},m_{Y_{1}})$-planes through Eq. (III.7) with the three choices of
the sine of the mixing angle $\sin{\alpha_{Y}}$ as $1/\sqrt{2}$, $0.3$ and
$0.2$.333Note that the sign of $\sin{\alpha_{Y}}$ is not important. We can
compensate a negative sign by flipping the sign of the coupling
$\lambda_{u}^{32}$ or $\lambda_{d}^{32}$. Here, $m_{V}$ is set as
$m_{V}=6\,m_{Y_{2}}$ and $m_{Y_{1}}$ is formulated as $m_{Y_{1}}=a\,m_{Y_{2}}$
by use of the factor $a$, where the range $[1.1,6.0]$ is considered in the
three plots. Note that in the case that $m_{Y_{1}}$ and $m_{Y_{2}}$ are
completely degenerated, the two contributions being proportional to $m_{b}$
are exactly canceled out between them and no cancellation mechanism works in
$\tau^{-}\to\mu^{-}\gamma$. Here, almost all the shown regions in Fig. 2
(where $\sin{\alpha_{Y}}$ is greater than $0.2$), the target values of the
ratio $(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}$ are greater than $0.05$,
which means that we can adjust naturally the two couplings for realizing the
cancellation. However, as we will see in the following section, when the ratio
$(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}$ gets to be small, it is hard to
explain the excess of $h\to\mu\tau$.

The Wilson coefficient $C_{R}^{\gamma}$ can be rewritten in terms of the ratio
in (III.7), which we will define as
$({\lambda_{u}^{32*}}/{\lambda_{d}^{32*}})_{\rm cancel}$,

$\displaystyle
C_{R}^{\gamma}\simeq\frac{N_{c}e}{32\pi^{2}m_{V}^{2}}\lambda_{u}^{32*}\lambda_{e}^{33*}m_{t}\Big{(}\frac{2}{3}I_{2}(x)+\frac{5}{3}J_{2}(x)\Big{)}\Bigg{[}1-\frac{\lambda_{d}^{32*}}{\lambda_{u}^{32*}}\left(\frac{\lambda_{u}^{32*}}{\lambda_{d}^{32*}}\right)_{\rm
cancel}\Bigg{]}.$ (III.8)

This equation shows again that, if
${\lambda_{d}^{32*}}/{\lambda_{u}^{32*}}=({\lambda_{d}^{32*}}/{\lambda_{u}^{32*}})_{\rm
cancel}$, $C_{R}^{\gamma}=0$. We can consider a deviation from the exact
cancellation by introducing $\delta$ in such a way that
${\lambda_{d}^{32*}}/{\lambda_{u}^{32*}}=({\lambda_{d}^{32*}}/{\lambda_{u}^{32*}})_{\rm
cancel}(1-\delta)$. Then we can take $\delta$ as a degree of required tuning
for cancellation in $\tau^{-}\to\mu^{-}\gamma$. In Fig. 3, the black lines
show a constant contour plot of $\delta$ in $(m_{V},\lambda_{\rm
conv}(\equiv|\lambda_{u}^{32}\lambda_{e}^{33}|))$-plane in percentage terms
when we take the upper limit on $C_{R}^{\gamma}$ (with $C_{L}^{\gamma}=0$) in
Eq. (III.6). The plot shows that we need fine-tuning at the level of $0.1\%$
to explain the excess of $h\to\mu\tau$ consistently.

Figure 3: The blue region shows the 2$\sigma$ favored region for $h\to\mu\tau$
in $(m_{V},\lambda_{\rm conv})$-plane, where $\lambda_{\rm
conv}\equiv|\lambda_{u}^{32}\lambda_{e}^{33}|$. The black contours indicate
degrees of the fine tuning defined around Eq. (III.8) in percentage terms.

##  IV $h\to\mu\tau$

Figure 4: Feynman diagrams of one-loop correction for $H^{0}-\mu-\tau$ vertex.

The lepton flavor violating Higgs decay is evaluated from the Feynman diagrams
shown in Fig. 4. The divergence in diagram Fig. 4 (a) cancels those in Fig. 4
(c), (d), and the total result is finite, generating the dimension-four
effective operators

$\displaystyle{\cal H}_{\rm eff}(h\to\mu\tau)$ $\displaystyle=$ $\displaystyle
h\overline{\mu}(C_{R}P_{R}+C_{L}P_{L})\tau+{H.c}.$ (IV.1)

The dimensionless effective couplings $C_{R,L}$ are calculated to be

$\displaystyle C_{R}$ $\displaystyle=$
$\displaystyle-\lambda_{u}^{32*}\lambda_{e}^{33*}\frac{N_{c}m_{t}}{16\pi^{2}v}\Bigg{[}I_{acd}(r_{t},r_{h})+I_{a}(r_{t},r_{h})-{\lambda_{HR}}\frac{v^{2}}{m_{V}^{2}}I_{b}(r_{t},r_{h})\Bigg{]}$
(IV.2)
$\displaystyle-\lambda_{d}^{32*}\lambda_{e}^{33*}\frac{N_{c}m_{b}}{16\pi^{2}v}\Bigg{\\{}\sum_{i=1}^{2}O_{2i}O_{1i}\Big{[}I_{abc}(0,s_{i})+I_{a}(0,s_{i})\Big{]}$
$\displaystyle-\sum_{i,j=1}^{2}O_{2i}O_{1j}(O^{T}\Lambda
O)_{ji}\frac{v^{2}}{m_{Y_{b}}^{2}}I_{b2}(s_{ij},s_{j}){\Bigg{\\}}},$
$\displaystyle{C_{L}}$ $\displaystyle=$
$\displaystyle-{\lambda_{e}^{23}\lambda_{u}^{33}}\frac{N_{c}m_{t}}{16\pi^{2}v}\Bigg{[}I_{acd}(r_{t},r_{h})+I_{a}(r_{t},r_{h})-\lambda_{HX}\frac{v^{2}}{m_{V}^{2}}I_{b}(r_{t},r_{h})\Bigg{]}$
(IV.3)
$\displaystyle-{\lambda_{e}^{23}\lambda_{d}^{33}}\frac{N_{c}m_{b}}{16\pi^{2}v}\Bigg{\\{}\sum_{i=1}^{2}O_{2i}O_{1i}\Big{[}I_{abc}(0,s_{i})+I_{a}(0,s_{i})\Big{]}$
$\displaystyle-\sum_{i,j=1}^{2}O_{2i}O_{1j}(O^{T}\Lambda
O)_{ji}\frac{v^{2}}{m_{Y_{b}}^{2}}I_{b2}(s_{ij},s_{j}){\Bigg{\\}}},$

with $r_{t}=m_{t}^{2}/m_{V}^{2},r_{h}=m_{h}^{2}/m_{V}^{2}$,
$s_{i}=m_{h}^{2}/m_{Y_{i}}^{2}$, $s_{ij}=m_{Y_{i}}^{2}/m_{Y_{j}}^{2}$, and

$\displaystyle\Lambda\equiv\left(\begin{array}[]{cc}\lambda_{HR}&\lambda_{\rm
mix}\\\ \lambda_{\rm mix}&\widetilde{\lambda}_{HR}\\\ \end{array}\right).$
(IV.6)

Note that the coupling combinations, $\lambda_{u}^{32*}\lambda_{e}^{33*}$ and
$\lambda_{d}^{32*}\lambda_{e}^{33*}$ in $C_{R}$;
$\lambda_{e}^{23}\lambda_{u}^{33}$ and $\lambda_{e}^{23}\lambda_{d}^{33}$ in
$C_{L}$, are also found in the terms in $C_{R}^{\gamma}$ and $C_{L}^{\gamma}$
for describing primary contributions to $\tau^{-}\to\mu^{-}\gamma$,
respectively. But here, no sizable cancellation emerges between terms being
proportional to $m_{t}$ and $m_{b}$ when we adjust parameters for realizing
the cancellation in $\tau^{-}\to\mu^{-}\gamma$. We ignore the apparently
irrelevant terms being proportional to $m_{\tau}$ or $m_{\mu}$, which arise
from chirality flips in the external lines. The loop functions are

$\displaystyle I_{acd}(r_{t},r_{h})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}-2\int[dx]\log\Big{[}x_{3}+(1-x_{3})r_{t}-x_{1}x_{2}r_{h}-i\varepsilon\Big{]}$
$\displaystyle+\int_{0}^{1}dx\log\Big{[}x+(1-x)r_{t}-i\varepsilon\Big{]},$
$\displaystyle I_{a}(r_{t},r_{h})$ $\displaystyle=$
$\displaystyle\int[dx]\frac{x_{1}x_{2}r_{h}-r_{t}}{x_{3}+(1-x_{3})r_{t}-x_{1}x_{2}r_{h}},$
$\displaystyle I_{b}(r_{t},r_{h})$ $\displaystyle=$
$\displaystyle\int[dx]\frac{1}{1-x_{3}+x_{3}r_{t}-x_{1}x_{2}r_{h}},$
$\displaystyle I_{b2}(s_{ij},s_{j})$ $\displaystyle=$
$\displaystyle\int[dx]\frac{1}{x_{1}s_{ij}+x_{2}-x_{1}x_{2}s_{j}},$ (IV.7)

where
$\int[dx]\equiv\int_{0}^{1}dx_{1}\int_{0}^{1}dx_{2}\int_{0}^{1}dx_{3}\,\delta(1-x_{1}-x_{2}-x_{3})$
and $\varepsilon$ represents an infinitesimal positive value.444When $X$ is
real, the relation $\log\left[X\pm
i\varepsilon\right]=\log\left[\left|X\right|\right]\pm
i\pi\theta\left(-X\right)$ with the Heaviside theta function $\theta$ is
useful. The form of the partial width $\Gamma_{h\to\mu^{-}\tau^{+}}$ is
described by use of the Wilson coefficient $C_{R}$ and $C_{L}$ in Eq. (IV.1)
as

$\displaystyle\Gamma_{h\to\mu^{-}\tau^{+}}=\frac{\bar{\beta}}{16\pi
m_{h}}\left[{(m_{h}^{2}-m_{\mu}^{2}-m_{\tau}^{2})}\left(|C_{R}|^{2}+|C_{L}|^{2}\right)-2m_{\mu}m_{\tau}\left(C_{R}C_{L}^{\ast}+C_{L}C_{R}^{\ast}\right)\right],$
(IV.8)

with the kinetic factor

$\displaystyle\bar{\beta}=\sqrt{1-\frac{2(m_{\mu}^{2}+m_{\tau}^{2})}{m_{h}^{2}}+\frac{(m_{\mu}^{2}-m_{\tau}^{2})^{2}}{m_{h}^{4}}},$
(IV.9)

while that of the conjugated process $\Gamma_{h\to\mu^{+}\tau^{-}}$ is
straightforwardly obtained by the replacements $C_{R}\to C_{R}^{\ast}$ and
$C_{L}\to C_{L}^{\ast}$. The inclusive width $\Gamma_{h\to\mu\tau}$ is simply
defined as

$\displaystyle\Gamma_{h\to\mu\tau}=\Gamma_{h\to\mu^{-}\tau^{+}}+\Gamma_{h\to\mu^{+}\tau^{-}}.$
(IV.10)

We use the value $\Gamma^{\text{SM}}_{h}=4.07\,\text{MeV}$ in
$m_{h}=125\,\text{GeV}$ reported by the LHC Higgs Cross Section Working Group
Heinemeyer:2013tqa  for evaluating $\mathcal{B}(h\to\mu\tau)$ in our model.

In the following analysis, as we did in the $\tau^{-}\to\mu^{-}\gamma$ in Sec.
III, we adopt the assumption of $C_{L}=0$. Among many terms in (IV.2), the two
terms in the first line, i.e., the top-quark contribution in Fig. 4 (a)
dominates and we ignore the bottom-quark contributions in the following
numerical estimation. In Fig. 3, we show the $2\sigma$ range to explain the
excess in $h\to\mu\tau$ shown in Eq. (I.1) in $(m_{V},\lambda_{\rm
conv})$-plane, where $\lambda_{\rm
conv}\equiv|\lambda_{u}^{32}\lambda_{e}^{33}|$. We set ${\lambda_{HR}}=1$,
which is the coupling of the subleading term in Eq. (IV.2) with the
suppression factor $v^{2}/m_{V}^{2}$. Here, an upper limit on
$\lambda_{\text{conv}}$ is estimated as
$(\lambda_{u}^{32}\lambda_{e}^{33})|_{\text{max}}=((\lambda_{u}^{32}/\lambda_{d}^{32})\lambda_{d}^{32}\lambda_{e}^{33})|_{\text{max}}\simeq
0.25\cdot 4\pi\cdot 4\pi\simeq 40$, where $0.25$ means a typical maximal value
of the ratio $(\lambda_{u}^{32}/\lambda_{d}^{32})$ shown in Fig. 2 (when
$m_{V}$ is multi TeV and $\sin{\alpha_{Y}}=1/\sqrt{2}$), and $4\pi$ comes from
perturbative regime in $\lambda_{d}^{32}$ and $\lambda_{e}^{33}$.

Combining Fig. 2 and Fig. 3, we can see that it is possible to explain the
excess shown in Eq. (I.1) in our scenario. At first, we will remember the
relation in the LQ’s masses, $m_{Y_{i}}\sim m_{V}/6$ for ensuring natural
cancellations between $\lambda^{32}_{u}$ and $\lambda^{32}_{d}$ in
$\tau^{-}\to\mu^{-}\gamma$. When we request the (exact) cancellation in
$\tau^{-}\to\mu^{-}\gamma$, as shown in Fig. 2, the ratio
$(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}$ should be smaller than unity.
Considering a typical scale of $m_{V}$ is more than a few TeV through the
relation $m_{Y_{i}}\sim m_{V}/6$ and the latest LHC bounds on $m_{\text{LQ}}$,
as a rough estimation, $\lambda_{\text{conv}}$ needs to be larger than around
ten. Taking into account the bound via perturbativity $\lambda_{e}^{33}<4\pi$,
roughly speaking, $\lambda_{u}^{32}$ should be greater than one through the
definition of $\lambda_{\text{conv}}$. Following this property, we should
think about the property of the ratio
$(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}$. Roughly, greater than $0.1$ is
required for realizing the above inequality ${\lambda_{u}^{32}}>1$ within the
region where $\lambda_{d}^{32}$ is still perturbative
($\lambda_{d}^{32}<4\pi$). This means that the mixing angle $\alpha_{Y}$
should be large to some extent since when $\alpha_{Y}$ becomes far from the
maximal case, the region with $(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}>0.1$
shrinks or disappears.555It is possible to modify the mass relation
$m_{Y_{i}}\sim m_{V}/6$ without caring about the difference between
$\lambda_{u}^{32}$ and $\lambda_{d}^{32}$. When $m_{V}$ is heavier than the
case following $m_{Y_{i}}\sim m_{V}/6$, the top contribution in
$\tau^{-}\to\mu^{-}\gamma$ decreases and the ratio
$(\lambda_{u}^{32}/\lambda_{d}^{32})^{\ast}$ can get to be large, which means
that larger ${\lambda_{u}^{32}}$ would be realizable. On the other hand,
however, a large $m_{V}$ suppresses the process $h\to\mu\tau$.

As an example, we can satisfy $\tau^{-}\to\mu^{-}\gamma$ constraint, with
$(m_{V},m_{Y_{1}},m_{Y_{2}})=(3.6,0.9,0.6)\,\text{TeV}$ and
$\sin{\alpha_{Y}}=1/\sqrt{2}$ leading to
$\lambda_{u}^{32}/\lambda_{d}^{32}\approx 0.15$. If we take
$\lambda_{d}^{32}\approx 10$ and $\lambda_{e}^{33}\approx 4$, we get
$\lambda_{\rm conv}\approx 6$, which can explain the central value shown in
Eq. (I.1).

##  V $(g-2)_{\mu}$

The anomalous magnetic moment of the muon has been measured to 0.5 ppm level
Bennett:2004pv ,

$\displaystyle a_{\mu}^{\rm exp}=116\,592\,080(63)\times 10^{-11}.$ (V.1)

Theoretical calculation in the SM has similar precision Miller:2007kk

$\displaystyle a_{\mu}^{\rm SM}=116\,591\,785(61)\times 10^{-11}.$ (V.2)

The discrepancy

$\displaystyle\Delta a_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=(295\pm
88)\times 10^{-11}$ (V.3)

is believed to come from new physics contributions.

However, we should also keep in mind that there is a possibility that the
discrepancy (or part of it) comes from underestimated uncertainties in
hadronic part, for example, in hadronic light-by-light scattering. Lattice
calculations Blum:2014oka ; Blum:2015gfa ; Green:2015mva  as well as
calculations using dispersion relations Colangelo:2014dfa ; Pauk:2014rfa ;
Colangelo:2015ama  will reduce the hadronic uncertainties in the future.

In our model the leptoquark contribution to $(g-2)_{\mu}$ is given by

$\displaystyle\Delta a_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{N_{c}m_{\mu}}{8\pi^{2}m_{V}^{2}}\Bigg{[}m_{\mu}\Big{(}\left|\lambda_{e}^{23}\right|^{2}+\left|\lambda_{u}^{32}\right|^{2}\Big{)}\Big{(}\frac{2}{3}I_{1}(x)+\frac{5}{3}J_{1}(x)\Big{)}+{\rm
Re}(\lambda_{u}^{32}\lambda_{e}^{23})m_{t}\Big{(}\frac{2}{3}I_{2}(x)+\frac{5}{3}J_{2}(x)\Big{)}\Bigg{]}$
(V.4) $\displaystyle-$
$\displaystyle\sum_{j=1,2}\frac{N_{c}m_{\mu}}{8\pi^{2}m_{Y_{j}}^{2}}\Bigg{[}m_{\mu}\Big{(}\left|\lambda_{e}^{23}\right|^{2}O_{1j}^{2}+\left|\lambda_{d}^{32}\right|^{2}O_{2j}^{2}\Big{)}\Big{(}-\frac{1}{3}I_{1}(y_{j})+\frac{2}{3}J_{1}(y_{j})\Big{)}$
$\displaystyle+$ $\displaystyle{\rm
Re}(\lambda_{d}^{32}\lambda_{e}^{23})O_{1j}O_{2j}m_{b}\Big{(}-\frac{1}{3}I_{2}(y_{j})+\frac{2}{3}J_{2}(y_{j})\Big{)}\Bigg{]},$

the loop functions are given in (III.3). We notice that, if we set
$m_{\mu,\tau}\to 0$ in $C_{R}^{\gamma}$ in Eq. (III.2) and inside the square
brackets in $\Delta a_{\mu}$ in Eq. (V.4), $\Delta a_{\mu}$ is exactly
proportional to $C_{R}^{\gamma}$ as

$\displaystyle\Delta
a_{\mu}=-\frac{4m_{\mu}}{e}\frac{\lambda_{e}^{23}}{\lambda_{e}^{33}}C_{R}^{\gamma},$
(V.5)

where we assumed all the couplings are real. If we use the current upper bound
of $C_{R}^{\gamma}$ in (III.6), we get

$\displaystyle\Delta a_{\mu}\approx-(66.3\times
10^{-11})\frac{\lambda_{e}^{23}}{\lambda_{e}^{33}}.$ (V.6)

Therefore we see that, if
$-7\,\lesssim\,{\lambda_{e}^{23}}/{\lambda_{e}^{33}}\,\lesssim\,-2$, we can
explain the muon $(g-2)_{\mu}$ with $\pm 2\sigma$ accuracy.

Figure 5: A two-loop Barr-Zee type diagram for $(g-2)_{\mu}$.

Since in our case the Yukawa couplings to explain $h\to\mu\tau$ and
$(g-2)_{\mu}$ are rather large, one may expect higher order diagrams such as
Barr-Zee type two-loop diagrams Barr:1990vd  may enhance $(g-2)_{\mu}$ as in
the case of MSSM with large $\tan\beta$ Arhrib:2001xx ; Baek:2004tm . In our
estimate, the dominant two-loop diagram is shown in Fig. 5. Other diagrams,
such as the one with LQs running inside the loop, are suppressed, for example,
by small muon mass, and we do not consider them. Although the diagram in Fig.
5 may look comparable with the one-loop diagrams due to large
$\lambda_{u}^{32}\lambda_{e}^{23}$, we still need chirality flip inside the
muon line in the fermionic triangle loop. Concretely, the diagram is estimated
to be suppressed at least by

$\displaystyle\sim{1\over
16\pi^{2}}\lambda_{u}^{32}\lambda_{e}^{33}\frac{m_{\mu}}{m_{t}}\sim 10^{-3}$
(V.7)

compared to the one-loop diagram.

##  VI Conclusion

In this paper, we considered the recent CMS excess in $h\to\mu\tau$ and the
muon $(g-2)$ anomaly. We showed that we can accommodate both discrepancies by
introducing two leptoquarks $R_{2}(3,2,7/6)$ and $\widetilde{R}_{2}(3,2,1/6)$
that are free from proton decay problems at renormalizable level. The
constraints from lepton flavor violating process $\tau^{-}\to\mu^{-}\gamma$
can be evaded by a natural cancellation between leptoquark contributions with
some tuning on $\lambda_{u}^{32}$ and $\lambda_{d}^{32}$, where their orders
can be the same. When the cancellation is realized, sizable couplings
contributing to $h\to\mu\tau$ are allowed and then we give a reasonable
explanation on the excess. The $(g-2)_{\mu}$ anomaly is also explained.
Finally, we mention that various kinds of other anomalies in flavor physics
have been reported BaBar_tau ; BaBar_tau13 ; LHCb_Kstar ; LHCb ; LHC_tau ;
Belle_tau . Giving a more exhaustive explanation in the context of leptoquarks
would be an important task B-leptoquark .

###### Acknowledgements.

K.N. is grateful to Tomohiro Abe, Shinya Kanemura, Kin-ya Oda, Makoto Sakamoto
and Koji Tsumura for fruitful discussions. This work is supported in part by
National Research Foundation of Korea (NRF) Research Grant No.
NRF-2015R1A2A1A05001869 (S.B.).

## References

  * (1) W. Buchmuller, R. Ruckl and D. Wyler, Phys. Lett. B 191 (1987) 442 [Phys. Lett. B 448 (1999) 320].
  * (2) G. Aad et al. [ATLAS Collaboration], arXiv:1508.04735 [hep-ex].
  * (3) V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B 739, 229 (2014) [arXiv:1408.0806 [hep-ex]].
  * (4) V. Khachatryan et al. [CMS Collaboration], JHEP 1507, 042 (2015) [arXiv:1503.09049 [hep-ex]].
  * (5) CMS Collaboration, Report No. PAS EXO-12-041, http://cds.cern.ch/record/1742179/files/.
  * (6) F. S. Queiroz, K. Sinha and A. Strumia, Phys. Rev. D 91 (2015) 3, 035006 [arXiv:1409.6301 [hep-ph]].
  * (7) V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B 749, 337 (2015) [arXiv:1502.07400 [hep-ex]].
  * (8) G. Aad et al. [ATLAS Collaboration], arXiv:1508.03372 [hep-ex].
  * (9) G. Blankenburg, J. Ellis and G. Isidori, Phys. Lett. B 712 (2012) 386 [arXiv:1202.5704 [hep-ph]].
  * (10) R. Harnik, J. Kopp and J. Zupan, JHEP 1303 (2013) 026 [arXiv:1209.1397 [hep-ph]].
  * (11) S. Davidson and P. Verdier, Phys. Rev. D 86 (2012) 111701 [arXiv:1211.1248 [hep-ph]].
  * (12) J. Kopp and M. Nardecchia, JHEP 1410 (2014) 156 [arXiv:1406.5303 [hep-ph]].
  * (13) A. Celis, V. Cirigliano and E. Passemar, arXiv:1409.4439 [hep-ph].
  * (14) C. J. Lee and J. Tandean, JHEP 1504, 174 (2015) [arXiv:1410.6803 [hep-ph]].
  * (15) L. de Lima, C. S. Machado, R. D. Matheus and L. A. F. do Prado, arXiv:1501.06923 [hep-ph].
  * (16) I. de Medeiros Varzielas and G. Hiller, JHEP 1506, 072 (2015) [arXiv:1503.01084 [hep-ph]].
  * (17) B. Bhattacherjee, S. Chakraborty and S. Mukherjee, arXiv:1505.02688 [hep-ph].
  * (18) Y. n. Mao and S. h. Zhu, arXiv:1505.07668 [hep-ph].
  * (19) X. G. He, J. Tandean and Y. J. Zheng, JHEP 1509, 093 (2015) [arXiv:1507.02673 [hep-ph]].
  * (20) T. Goto, R. Kitano and S. Mori, arXiv:1507.03234 [hep-ph].
  * (21) W. Altmannshofer, S. Gori, A. L. Kagan, L. Silvestrini and J. Zupan, arXiv:1507.07927 [hep-ph].
  * (22) A. Pilaftsis, Phys. Lett. B 285, 68 (1992).
  * (23) A. Arhrib, Y. Cheng and O. C. W. Kong, Europhys. Lett. 101 (2013) 31003 [arXiv:1208.4669 [hep-ph]].
  * (24) A. Falkowski, D. M. Straub and A. Vicente, JHEP 1405 (2014) 092 [arXiv:1312.5329 [hep-ph]].
  * (25) A. Dery, A. Efrati, Y. Nir, Y. Soreq and V. Susič, Phys. Rev. D 90 (2014) 115022 [arXiv:1408.1371 [hep-ph]].
  * (26) M. D. Campos, A. E. C. Hernández, H. Päs and E. Schumacher, Phys. Rev. D 91 (2015) 11, 116011 [arXiv:1408.1652 [hep-ph]].
  * (27) D. Aristizabal Sierra and A. Vicente, Phys. Rev. D 90 (2014) 11, 115004 [arXiv:1409.7690 [hep-ph]].
  * (28) J. Heeck, M. Holthausen, W. Rodejohann and Y. Shimizu, Nucl. Phys. B 896, 281 (2015) [arXiv:1412.3671 [hep-ph]].
  * (29) A. Crivellin, G. D’Ambrosio and J. Heeck, Phys. Rev. Lett. 114, 151801 (2015) [arXiv:1501.00993 [hep-ph]].
  * (30) I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik, N. Košnik and I. Nišandžic, JHEP 1506 (2015) 108 [arXiv:1502.07784 [hep-ph]].
  * (31) Y. Omura, E. Senaha and K. Tobe, JHEP 1505, 028 (2015) [arXiv:1502.07824 [hep-ph]].
  * (32) A. Crivellin, G. D’Ambrosio and J. Heeck, Phys. Rev. D 91, no. 7, 075006 (2015) [arXiv:1503.03477 [hep-ph]].
  * (33) D. Das and A. Kundu, Phys. Rev. D 92, no. 1, 015009 (2015) [arXiv:1504.01125 [hep-ph]].
  * (34) C. W. Chiang, H. Fukuda, M. Takeuchi and T. T. Yanagida, arXiv:1507.04354 [hep-ph].
  * (35) A. Crivellin, J. Heeck and P. Stoffer, arXiv:1507.07567 [hep-ph].
  * (36) K. Cheung, W. Y. Keung and P. Y. Tseng, arXiv:1508.01897 [hep-ph].
  * (37) F. J. Botella, G. C. Branco, M. Nebot and M. N. Rebelo, arXiv:1508.05101 [hep-ph].
  * (38) S. Baek and Z. F. Kang, arXiv:1510.00100 [hep-ph].
  * (39) F. Jegerlehner and A. Nyffeler, Phys. Rept. 477 (2009) 1 [arXiv:0902.3360 [hep-ph]].
  * (40) M. Benayoun, P. David, L. DelBuono and F. Jegerlehner, Eur. Phys. J. C 73 (2013) 2453 [arXiv:1210.7184 [hep-ph]].
  * (41) J. M. Arnold, B. Fornal and M. B. Wise, Phys. Rev. D 88, 035009 (2013) [arXiv:1304.6119 [hep-ph]].
  * (42) B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 104, 021802 (2010) [arXiv:0908.2381 [hep-ex]].
  * (43) S. Heinemeyer et al. [LHC Higgs Cross Section Working Group Collaboration], arXiv:1307.1347 [hep-ph].
  * (44) G. W. Bennett et al. [Muon g-2 Collaboration], Phys. Rev. Lett. 92 (2004) 161802 [hep-ex/0401008].
  * (45) J. P. Miller, E. de Rafael and B. L. Roberts, Rept. Prog. Phys. 70 (2007) 795 [hep-ph/0703049].
  * (46) T. Blum, S. Chowdhury, M. Hayakawa and T. Izubuchi, Phys. Rev. Lett. 114, no. 1, 012001 (2015) doi:10.1103/PhysRevLett.114.012001 [arXiv:1407.2923 [hep-lat]].
  * (47) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin and C. Lehner, arXiv:1510.07100 [hep-lat].
  * (48) J. Green, N. Asmussen, O. Gryniuk, G. von Hippel, H. B. Meyer, A. Nyffeler and V. Pascalutsa, arXiv:1510.08384 [hep-lat].
  * (49) G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, JHEP 1409, 091 (2014) doi:10.1007/JHEP09(2014)091 [arXiv:1402.7081 [hep-ph]].
  * (50) V. Pauk and M. Vanderhaeghen, Phys. Rev. D 90, no. 11, 113012 (2014) doi:10.1103/PhysRevD.90.113012 [arXiv:1409.0819 [hep-ph]].
  * (51) G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, JHEP 1509, 074 (2015) doi:10.1007/JHEP09(2015)074 [arXiv:1506.01386 [hep-ph]].
  * (52) S. M. Barr and A. Zee, Phys. Rev. Lett. 65, 21 (1990) [Phys. Rev. Lett. 65, 2920 (1990)]. doi:10.1103/PhysRevLett.65.21
  * (53) A. Arhrib and S. Baek, Phys. Rev. D 65, 075002 (2002) doi:10.1103/PhysRevD.65.075002 [hep-ph/0104225].
  * (54) S. Baek, Phys. Lett. B 595, 461 (2004) doi:10.1016/j.physletb.2004.05.071 [hep-ph/0406007].
  * (55) J. P. Lees et al. [BaBar Collaboration], Phys. Rev. Lett. 109 (2012) 101802 [arXiv:1205.5442 [hep-ex]].
  * (56) J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 88 (2013) 7, 072012 [arXiv:1303.0571 [hep-ex]].
  * (57) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 111 (2013) 191801 [arXiv:1308.1707 [hep-ex]].
  * (58) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 113 (2014) 151601 [arXiv:1406.6482 [hep-ex]].
  * (59) R. Aaij et al. [LHCb Collaboration], arXiv:1506.08614 [hep-ex].
  * (60) M. Huschle et al. [Belle Collaboration], arXiv:1507.03233 [hep-ex].
  * (61) S. Baek and K. Nishiwaki, work in progress.