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### Algebra Question 5e

5. Solve the following inequalities and sketch the sketch the graphs on the number line.

e)

5.  Solve the following inequalities and sketch the sketch the graphs on the number line.

e)

e)

5. Solve the following inequalities and sketch the sketch the graphs on the number line.

e)

Solution:

e)      i.e.        or

That is     or     i.e.

The Real Number Line

The point on the number line which corresponds to a given number is called the graph of the number.

Notation
The symbol denotes is greater than;
The symbol denotes is less than;
The symbol denotes is greater than or equal to;
The symbol denotes is less than or equal to.


text:
# Unbiased Estimates in Policy Gradient

## Introduction

In reinforcement learning policy gradient methods, the goal is usually to maximize $\mathbb{E}[R_t]$, the expected value of return $R$ at time step $t$, where $R_t = \sum_{k=0}^{\infty} \gamma^k r_{t+k}$.

We have parameterized the policy $\pi(a|s; \theta)$ and update the parameter $\theta$ so that $\mathbb{E}[R_t]$ is maximized.

Standard REINFORCE algorithm updates policy in the direction $\nabla_{\theta} R_t \log \pi (a_t|s_t; \theta)$. That is to say, we have a loss function of $-\log R_t \pi (a_t|s_t; \theta)$ and we are minimizing the loss.

But in order to mamixize $\mathbb{E}[R_t]$, why do we have to calculate the the derivatives for $R_t \log \pi (a_t|s_t; \theta)$? It turns out that it is because $\nabla_{\theta} R_t \log \pi (a_t|s_t; \theta)$ is an unbiased estimate of $\nabla_{\theta} \mathbb{E}[R_t]$, i.e., $\mathbb{E}[R_t \nabla_{\theta} \log \pi (a_t|s_t; \theta)] = \nabla_{\theta} \mathbb{E}[R_t]$.

## Proof

We use the definition of expected value, because random variable $R_t$ is dependent on $a_t$ and it follows distribution $\pi (a_t|s_t; \theta)$.

$$\nabla_{\theta} \mathbb{E}[R_t] = \nabla_{\theta} \int R_t \pi (a_t|s_t; \theta) d(a_t)$$

Then we apply a special case of Leibniz Integral Rule, where

$$\frac{d}{dx} \int_{a}^{b} f(x,t) dt = \int_{a}^{b} \frac{\partial}{\partial x} f(x,t) dt$$

We then have

$$\nabla_{\theta} \int R_t \pi (a_t|s_t; \theta) d(a_t) = \int \nabla_{\theta} [ R_t \pi (a_t|s_t; \theta) ] d(a_t)$$

Because $R_t$ is dependent on $a_t$ but not dependent on $\theta$ (Is it?). We have

$$\int \nabla_{\theta} [ R_t \pi (a_t|s_t; \theta) ] d(a_t) = \int R_t \nabla_{\theta} \pi (a_t|s_t; \theta) d(a_t)$$

We apply an identity trick here

\begin{aligned} \int R_t \nabla_{\theta} \pi (a_t|s_t; \theta) d(a_t) &= \int R_t \pi (a_t|s_t; \theta) \frac{\nabla_{\theta} \pi (a_t|s_t; \theta)}{\pi (a_t|s_t; \theta)} d(a_t) \\ &= \int R_t \pi (a_t|s_t; \theta) \nabla_{\theta} \log \pi (a_t|s_t; \theta) d(a_t) \end{aligned}

Because the random variable $R_t \nabla_{\theta} \log \pi (a_t|s_t; \theta)$ also follows distribution $\pi (a_t|s_t; \theta)$, we have

$$\int R_t \pi (a_t|s_t; \theta) \nabla_{\theta} \log \pi (a_t|s_t; \theta) d(a_t) = \mathbb{E}[R_t \nabla_{\theta} \log \pi (a_t|s_t; \theta)]$$

Therefore,

$$\nabla_{\theta} \mathbb{E}[R_t] = \mathbb{E}[R_t \nabla_{\theta} \log \pi (a_t|s_t; \theta)]$$

This concludes the proof.

## Generalization

More generally it is not hard to see that,

$$\nabla_{\theta} \mathbb{E}[f(z)] = \mathbb{E}[f(z) \nabla_{\theta} \log p (z; \theta)]$$

## Extension

We also use REINFORCE with baseline to reduce the variance of the estimate of $\nabla_{\theta} \mathbb{E}[R_t]$ while keeping the estimate unbiased.

In stead of using $\nabla_{\theta} R_t \log \pi (a_t|s_t; \theta)$ as an unbiased estimate of $\nabla_{\theta} \mathbb{E}[R_t]$, we use $\nabla_{\theta} (R_t - v(s_t)) \log \pi (a_t|s_t; \theta)$ as long as $v(s_t)$ does not dependent on $a$.

To see why $\mathbb{E}[(R_t - v(s_t)) \nabla_{\theta} \log \pi (a_t|s_t; \theta)] = \nabla_{\theta} \mathbb{E}[R_t]$, we have

$$\nabla_{\theta} \mathbb{E}[R_t] = \nabla_{\theta} \int R_t \pi (a_t|s_t; \theta) d(a_t) = \nabla_{\theta} \int R_t \pi (a_t|s_t; \theta) d(a_t) - \nabla_{\theta} \int v(s_t) \pi (a_t|s_t; \theta) d(a_t)$$

Where $\nabla_{\theta} \int v(s_t) \pi (a_t|s_t; \theta) d(a_t) = 0$.

Because $v$ use different parameters other than $\theta$ and it is not dependent on $a$,

$$\nabla_{\theta} \int v(s_t) \pi (a_t|s_t; \theta) d(a_t) = v(s_t) \nabla_{\theta} \int \pi (a_t|s_t; \theta) d(a_t) = v(s_t) \nabla_{\theta} 1 = v(s_t) \times 0 = 0$$

## Quick Proof to Leibniz Integral Rule

\begin{aligned} \frac{d}{dx} \int_{a}^{b} f(x,t) dt &= \frac{d}{dx} \int_{a}^{b} \int_{\mathbf{x}} \frac{\partial f(x,t)}{\partial x} dx dt\\ &= \frac{d}{dx} \int_{\mathbf{x}} \int_{a}^{b} \frac{\partial f(x,t)}{\partial x} dt dx\\ &= \frac{d}{dx} \int_{\mathbf{x}} (\int_{a}^{b} \frac{\partial f(x,t)}{\partial x} dt) dx\\ &= \int_{a}^{b} \frac{\partial f(x,t)}{\partial x} dt\\ \end{aligned}

Thanks to the guidance from my friend Guotu Li on this.

## Final Remarks

There is a statement or an assumption we made during the proof, that is $R_t$ is not dependent on $\theta$. It is because in this case $R_t$ is a scala value instead of an output from the policy neural network.

Lei Mao

03-15-2019

03-15-2019

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# Tag Info

21

As mentioned elsewhere on this site, Lo, Mamaysky, and Wang (2000) do exactly what you're talking about, namely algorithmic detection of head and shoulders patterns. Their definition: Head-and-shoulders (HS) and inverted head-and-shoulders (IHS) patterns are characterized by a sequence of five consecutive local extrema $E_1,...,E_5$ such that  HS ...

17

I've worked at a hedge fund that allowed GA-derived strategies. For safety, it required that all models be submitted long before production to make sure that they still worked in the backtests. So there could be a delay of up to several months before a model would be allowed to run. It's also helpful to separate the sample universe; use a random half of the ...

14

I think the biggest problem that genetic algorithms have are overfitting, data snooping bias and that they are black boxes (not so much like Neural Networks but still - it depends on the way they are implemented). I think they are not used very much. I guess there are a few hedge funds out there that use it but all in all they were hyped and then busted. ...

14

There are tons of languages used in this field. As for Java-based trading platforms, Marketcetera is popular with customers. To justify switching languages, you'd need to show that there is a bottleneck preventing your team from collecting more P&L. Have you run a profiler and compared the results with tcpdump? You must show that your existing platform ...

12

I would recommend that you read "Evidence-Based Technical Analysis" by David Aronson. Firstly, I am mentioning it because it is a highly worthwhile book. Secondly, on pp151--161 he attempts to "objectify subjective TA", using the head-and-shoulders pattern as an example.

12

Oracle hosted a Trading Applications Developer Workshop in New York on March 15th, 2011. The slides from each of the presentations are here. One of them covers java for Trading Applications, and it seemed to me that the biggest issue raised by the audience was garbage collection. The presentation talks about some configuration parameters that can limit ...

11

I've applied GA to all sorts of things. I had some success in the deterministic world where a pattern actually existed and I knew that some physical structure existed (seismic analysis, vibration analysis, inventory calcs, etc). After I found a GA model that behaved, the real work started....figuring out why it behaved. I also generated a lot of GA ...

10

In a typical HFT scenario (process incoming UDP quotes and send TCP order entry responses) well written Java can compete with C++ for pure speed. If you need more speed, look to improve the following: Your code (setup a good benchmark and then profile and tune) Your networking environment (low-latency switches, DMA NICs) Your architecture (are you doing ...

9

I have these posts favorited from stackoverflow. They might help you. High Frequency Trading What programming language(s) is algorithmic trading software written in? Why does a derivative trading position always require C++ knowledge? Jane Street Capital, a high-frequency market making firm, uses OCaml. Here are videos from the head programmer where he ...

8

For years, I performed this brute-force search daily on my universe of tradable stocks and futures. It is a waste of time. If your computer discovers that hog futures and MSFT are cointegrated, for example, do you really care? I would never trade that pair. There is no economic connection between hogs and Microsoft, so I must assume that the reported, small ...

8

There's a lot of people here talking about how GAs are empirical, don't have theoretical foundations, are black-boxes, and the like. I beg to differ! There's a whole branch of economics devoted to looking at markets in terms of evolutionary metaphors: Evolutionary Economics! I highly recommend the Dopfer book, The Evolutionary Foundations of Economics, ...

8

Assuming you avoid data-snooping bias and all the potential pitfalls of using the past to predict the future, trusting genetic algorithms to find the "right" solution pretty much boils down to the same bet you make when you actively manage a portfolio, whether quantitatively or discretionary. If you believe in market efficiency then increasing your ...

7

Measuring expected shortfall (also known as conditional value-at-risk) answers the simpler question of "what is my average expected loss at the i-th quantile?" given the empirical distribution of returns. A variation is value-at-risk which measures the loss at the i-th quantile. Arguably you could leave at this this and you have your answer. You probably ...

6

Theoretically, the answer to the question is yes, a correlation matrix for potential pairs trades can be computed in $O\left((n^2t)^{(\omega+\epsilon)/3}\right)$ time, for any $\epsilon > 0$, where $\omega < 2.38$ is the so-called exponent of matrix multiplication. However, these algorithms have a reputation for having a very large constant factor ...

6

The most basic strategy is beta-based quantiles. That is to say, you first control for losses on your individual stock versus overall market performance. (Your trading strategy may or may not wish to hedge away the market factor using, say, SPX futures). Then you choose a quantile, call it the 5th percentile, beyond which you consider a move to be ...

5

All things being equal, stocks with the highest bid-ask spread present the greatest opportunity for the market maker The size of the opportunity (i.e. revenue expectation) can be represented as Volume * Bid-Ask Spread. Your algorithm should rank-order that revenue expectation Stocks with high current market values will tend to have narrower spreads and be ...

5

I have worked with C++, java and C# implementing a google like search engine for DOD in along with many other software that require high performance (using low level memory mapping and named pipes/tcp). In my experience, you cannot match speed of C++ with the managed code. In managed code, every call to new, garbage collection, etc. requires multiple ...

4

You question is quite strange: so you do not want to use methods inspired by bioinfo and genetics (neural networks, GA, geometry of folding, etc) but methods that are used in these fields? In terms of modeling, the problematics in bioinfo and genetics are mainly: tree or graph matching (to build metrics in the space of molecules), like in SIGMA: a ...

4

I won't give you the answer delivered on a silver platter but hopefully the following will get your started: a) you need to define exactly over which look-back period you aim to derive the maximum drawdown. b) you need to keep track of the max price while you iterate the look-back window. c) you need to keep track of the min price SUBSEQUENT to any NEW ...

3

Zipline, the opensource python backtester, has a batch and iterative implementation for max drawdown. Here is the batch: https://github.com/quantopian/zipline/blob/master/zipline/finance/risk.py#L284 Here is the iterative: https://github.com/quantopian/zipline/blob/master/zipline/finance/risk.py#L578 disclosure: I'm one of the zipline maintainers

3

Mathematically speaking that is an impossible thing to do. There are simply too many variables and randomness that you cannot do it. Rather than analyzing competitors trade data; why don't you analyze the market. You can consider every bar; as a trade. If the bar was up; it means you went long and made a profit. If the bar was down; it means you went short ...

3

Dynamic Time Warping, recursive, time-delayed feedforward neural networks, wavelets, empirical mode decomposition, ..., there's plenty of it. BUT If you want my advice, don't go this way, I wasted too much time doing things like that. Neither big nor small players (profitably and consistently) trade this way and for a good reason. Technical analysis is a ...

2

Yes there is a way to find kth largest element in an unsorted list in linear time here. However, depending on what program you are using, implementing the algorithm might not increase performance. The built-in sort function is probably optimized in C, and hence is very fast.

2

This paper by R. Marschinski and H. Kantz may be able to help: "Learning the Optimal Trading Strategy". I have not read it but the authors have published other work which is quite innovative. Within a realistic model of the stockmarket, we derive the most successful trading strategy. We first identify the agent who has realized the largest percentual ...

1

Regarding Joshua's inspired answer, I'm still not sure how you guarantee that scaling gives you the exact high and low values. I suppose that you could simulate until you get a result that is close enough. But that could be hard when, e.g., Open is near High and far from Low. An alternative solution is to construct a Brownian Bridge between Open and High, ...

1

(After the clarification, this answer is no longer relevant) Expected maximum drawdown is going to be highly sensitive to your choice of SDE, and to your calibration of it. Therefore you should play with a variety of parameterizations to estimate your model error. So far as efficient computation goes, we can regard this as a payoff very similar to a ...

1

Agree with the previous post, that it is very difficult to do, if not impossible. Having said that, I once saw a presentation on a (closed) project which was trying to use neural networks to train and form trading patterns. The group had access to a large lot of trading data and the profit (or losses) made from the trades over a certain time period (like, a ...

1

I do recommend to you the chapter 8 of the Ernest Chan book "Quantitative Trading: How to Build Your Own Algorithmic Trading Business" The chapter's name is "Conclusion: Can Independent Traders Succeed? " The conclusion is that yes but I now cant remember very well the chapter...

1

Well pattern recognition and image processing is so developed these days. This is cutting edge in CS now and if we could identify cancer or brain tumor on a hazy image or a suspect face on an industry cam then recognizing head and shoulders on a chart is really really easy. Support Vector Machines or entropy come to mind but there is a myriad of ...

1

Regarding trading, it depends upon one's style and temperament. Don't rely solely on Aronson's book and his views and a phrase quoted by Andrew Lo in his study. The formula posted by Tal Fishman of Head and Shoulders as quoted by Lo, Mamaysky and Wang (2000) is not exhaustive. There is a lot of scope for further improvement. However, there are many studies ...

Only top voted, non community-wiki answers of a minimum length are eligible


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Production and decay of up-type and down-type new heavy quarks through anomalous interactions at the LHC

# Production and decay of up-type and down-type new heavy quarks through anomalous interactions at the LHC

İ. Turk Çakır Istanbul Aydın University, Application and Research Center for Advanced Studies, 34295, Istanbul, Turkey    S. Kuday Istanbul Aydın University, Application and Research Center for Advanced Studies, 34295, Istanbul, Turkey    O. Çakır Istanbul Aydın University, Application and Research Center for Advanced Studies, 34295, Istanbul, Turkey Ankara University, Department of Physics, 06100, Ankara, Turkey
###### Abstract

We study the process (where and ) through the anomalous interactions of the new heavy quarks at the LHC. Considering the present limits on the masses and mixings, the signatures of the heavy quark anomalous interactions are discussed and analysed at the LHC for the center of mass energy of TeV. An important sensitivity to anomalous couplings TeV, TeV, TeV and TeV, TeV, TeV for the mass of 750 GeV of the new heavy quarks and can be reached for an integrated luminosity of fb.

## I introduction

The standard model (SM) of the strong and electroweak interactions describes succesfully the phenomena of particle physics. However, there are many unanswered questions suggesting the SM to be an effective theory. In order to answer some of the problems with the SM, additional new fermions can be accommodated in many models beyond the SM (see Refs. Holdom2009 (), Atre2009 (), Atre2011 (), Chakdar2013 () and references therein). The new heavy quarks could also be produced in pairs at the LHC with center of mass energy of TeV. However, due to the expected smallness of the mixing between the new heavy quarks and known quarks, the decay modes can be quite different from the one relevant to charged weak interactions. A new symmetry beyond the SM is expected to explain the smallness of these mixings. The arguments given in Ref. Fritzsch1999 () for anomalous interactions of the top quark are more valid for the new heavy quarks and due to their expected larger masses than the top quark.

The ATLAS experiment ATLAS-3 () and CMS experiment CMS-5 () have searched for the fourth generation of quarks and set limits on the mass of GeV and GeV at TeV. The pair production of new heavy quarks have been searched by the ATLAS experiment Aad:2014efa (), ATLAS:2012qe () and the GeV mass limits are set at TeV. The CMS experiment have excluded masses below 557 GeVCMS-6 () . The vector-like quarks have been searched by the ATLAS experiment Aad:2011yn (), Aad:2012uu () and set bounds as GeV for charged current channel and 760 GeV for neutral current channel at TeV. The CMS experiment CMS-7 (), CMS-8 () have set the lower bounds on the mass of GeV at TeV. Some of the final states in the searches of new phenomena ATLAS2014 () and excited quarks Aad:2013rna () can also be considered in relation with the new heavy quarks.

The anomalous resonant productions of the fourth family quarks have been studied in Refs. Ciftci:2008tc (); Cakir2009 () at the LHC with TeV. sThe possible single productions of fourth generation quarks via anomalous interactions at Tevatron have also been studied in Refs. Arik:2002sg (); Sahin (). The parameter space for the mixing of the fourth generation quarks have been presented in Ref. Bobrowski (). The CP violating flavour changing neutral current processes of the fourth generation quarks have been analysed in Ref. Eilam2009 (), and the large mixing between fourth generation and first three generations have been excluded under the proposed fit conditions. Investigation of the parameter space favoured by the precision electroweak data have been performed for the fourth SM family fermions in Ref. OPUCEM ().

In this work, we present the analysis of anomalous productions and decays of new heavy quarks and at the LHC. We have performed the fast simulation for the signal and background. Any observations of the invariant mass peak in the range of GeV and excess in the events with the final states originating from and can be interpreted as the signal for the new heavy quarks and via the anomalous interactions.

## Ii Heavy Quarks Anomalous Interactıons

A general theory that has the standard model (SM) as its low energy limit can be written as a series in with operators obeying the required symmetries. The effective Lagrangian for the anomalous interactions among the heavy quarks ( or ), ordinary quarks , and the gauge bosons can be written explicitly:

 L=∑qi=u,c,tκqiγΛQqige¯t′σμνqiFμν+∑qi=u,c,tκqiz2Λgz¯t′σμνqiZμν+∑qi=u,c,tκqig2Λgs¯t′σμνλaqiGμνa
 +∑qi=d,s,bκqiγΛQqige¯¯b′σμνqiFμν+∑qi=d,s,bκqiz2Λgz¯¯b′σμνqiZμν+∑qi=d,s,bκqig2Λgs¯¯b′σμνλaqiGμνa+h.c. (1)

where , and are the field strength tensors of the gauge bosons; ; are the Gell-Mann matrices; is the electric charge of the quark (); , and are the electromagnetic, neutral weak and the strong coupling constants, respectively. , where is the weak mixing angle. is the anomalous coupling with photon; is for the boson, and is the coupling with gluon. Finally, is the cutoff scale for the new interactions.

## Iii Decay Widths and Branchıngs

For the decay channels where , we use the effective Lagrangian to calculate the anomalous decay widths

 (2)
 Γ(Q′→γq)=12(κqγΛ)2αeQ2qm3Q′λ0 (3)
 Γ(Q′→Zq)=116(κqZΛ)2αem3Q′sin2θWcos2θWλZ√λr (4)

with

 λ0=1−3m2q/m2Q′+3m4q/m4Q′−m6q/m6Q′ (5)
 λr=1+m4W/m4Q′+m4q/m4Q′−2m2W/m2Q′−2m2q/m2Q′−2m2Wm2q/m4Q′ (6)
 λZ=2−m2Z/m2Q′−4m2q/m2Q′+2m4q/m4Q′−6mqm2Z/m3Q′−m2Zm2t/m4Q′−m4Z/m4Q′ (7)

The anomalous decay widths in different channels are proportional to , and they are assumed to be dominant for TeV over the charged current channels. In this case, if we take all the anomalous coupling equal then the branching ratios will be nearly independent of . We have used three parametrizations sets entitled PI, PII and PIII. For the PI parametrization, we assume the constant value TeV , and PII has the parameters TeV with . For PIII we take the couplings TeV with the same value of . The index is the generation number.

Table 1 and Table 2 present the decay width and branching ratios of the new heavy quark through anomalous interactions for the parametrizations PI, PII and PIII, respectively. Taking the anomalous coupling TeV we calculate the decay width GeV and GeV for GeV and GeV, respectively. The branching into channel is the largest and branching into channel is the smallest for equal anomalous couplings with the parametrization PI. On the other hand, PII and PIII parametrizations give higher branching ratios into () than () channels due to factor in the parametrizations.

For the new heavy quark the decay witdh and branching ratios are presented in Table 3 and Table 4 for the parametrizations PI, PII and PIII, respectively. We calculate the decay width, by taking the anomalous coupling TeV, GeV and GeV for GeV and GeV, respectively. The branching for is the largest (30%) and its the smallest for (0.2%) channel for equal anomalous couplings with the parametrization PI. For PII and PIII parametrizations the branching ratios into () are larger than () channels. The and decay widths are about the same values for PII and PIII parametrizations.

## Iv The Cross Sectıons

In order to study the new heavy quark productions at the LHC, we have used effective anomalous interaction vertices and implemented these vertices into the CalcHEP package CalcHEP (). In all of the numerical calculations, the parton distribution function are set to the CTEQ6L parametrization CTEQ6L (). The new heavy quarks can be produced through its anomalous couplings to the ordinary quarks and neutral vector bosons as shown in Fig. 1.

Total cross sections for the productions of new heavy quarks and are given in Table 5 and Table 6 for the parametrizations PI, PII and PIII, at the center of mass energy of 8 TeV and 13 TeV. For an illustration, taking the mass of new heavy quarks as 700 GeV the cross section of production is calculated as 8.50 pb (10.03 pb) for the parametrization PIII at TeV. It can be seen from Table 5 and Table 6, the cross sections decreases while the mass of the new heavy quark increases. The cross section for production is larger than the production with a factor of 1.2-1.8 (0.7-1.0) for PI (PII and PIII) parametrization depending on the considered mass range at TeV. The general behaviours of the production cross sections depending on the mass of heavy quarks are presented in Fig. 2 and Fig. 3 for different parametrizations.

### iv.1 Analysis of the process pp→W+bV+X (V=g,Z,γ) for t′ signal

The signal process () includes the exchange both in the -channel and -channel. The -channel contribution to the signal process would appear itself as resonance around the mass value in the invariant mass. The -channel gives the non-resonant contribution. We consider that the boson decays into lepton+missing transverse momentum with the branching ratio and boson decays into dilepton with the branching . In our analyses, we consider the signal in the , and channels, where . However, if one takes the hadronic decays the signal will be enhanced by a factor of

We have obtained the cross sections by using the cuts pseudorapidity and transverse momentum GeV for jets and photon, in Table 7 (Table8, Table 9) for PI (PII, PIII) parametrizations, respectively. It appears from signal significance calculations that the optimized transverse momentum cut is >100 GeV for analyses.

The backgrounds for the final state (where photon, jet and boson) are given in Table 10. We apply the following cuts to the final state photon and jets as and GeV. For the background cross section estimates, we assume the efficiency for b-tagging to be , and the rejection ratios for quark jets and for light quark jets since they are assumed to be mistagged as -jets.

In order to find the discovery limits we use the statistical significance as

 SS=√2[(S+B)ln(1+SB)−S] (8)

where and are the numbers of the signal and background events, respectively. In Figs. 4- 6, the integrated luminosity required to reach significance for the signal of anomalous interactions is shown for parametrization PI, PII and PIII at the LHC with TeV. It is seen from these figures that the channel requires more integrated luminosity than the other channels. By requiring the signal significance , the contour plots of and mass of quark are presented in Fig. 7. The results show that one can discover the quark anomalous couplings down to 0.1 TeV in the channel for =750 GeV.

### iv.2 Analysis of the process pp→bV+X (V=g,Z,γ) for b′ signal

The signal process () includes the new heavy quark exchange both in the -channel and -channel. The - channel contributes to the signal process as resonance around the mass value in the invariant mass, while the - channel contributes to the non-resonant behaviour. For this process, we consider the leptonic decays of boson. In the analyses, we consider the signal to be , and .

We have obtained the cross sections by using the pseudorapidity cuts and transverse momentum cuts GeV for jets and photon, in Table 11 (Table 12, Table 13 for PI (PII, PIII) parametrizations, respectively. It appears from signal significance calculation that the optimized transverse momentum cut is >200 GeV for analyses.

The backgrounds for the final state (where photon, jet and boson) are given in 14. We apply the following cuts to the final state photon and jets as and GeV. It can be noted that the background cross section decreases as the cuts increases. We assume the efficiency for b-tagging to be , and the rejection ratios for quark jets and for light quark jets.