Towards model-independent approach to the analysis of interference effects in pair production of new heavy quarks

Abstract

Abstract

We propose a model independent approach for the analysis of interference effects in the process of QCD pair production of new heavy quarks of different species that decay into Standard Model particles, including decays via flavour changing neutral currents. By adopting as ansatz a simple analytical formula we show that one can accurately describe the interference between two different such particle pairs leading to the same final state using information about masses, total widths and couplings. A study of the effects on differential distributions is also performed showing that, when interference plays a relevant role, the distributions of the full process can be obtained by a simple rescaling of the distributions of either quark contributing to the interference term. We also present the range of validity of the analytical expression that we have found.

Published for SISSA by Springer

Received: January 20, 2014 Revised: June 22, 2014

Accepted: July 3, 2014

Published: July 28, 2014

D. Barducci,a,b A. Belyaev,a,b J. Blamey,a S. Moretti,a,b L. Panizzia,b and H. Pragera,c

aSchool of Physics and Astronomy, University of Southampton,

Higheld, Southampton SO17 1BJ, U.K.

bParticle Physics Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, U.K.

ccole Normale Suprieure de Lyon, Universit de Lyon, 46 alle dItalie, 69364 Lyon cedex 07, France

E-mail: [email protected], mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , [email protected], mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We propose a model independent approach for the analysis of interference e ects in the process of QCD pair production of new heavy quarks of di erent species that decay into Standard Model particles, including decays via avour changing neutral currents. By adopting as ansatz a simple analytical formula we show that one can accurately describe the interference between two di erent such particle pairs leading to the same nal state using information about masses, total widths and couplings. A study of the e ects on di erential distributions is also performed showing that, when interference plays a relevant role, the distributions of the full process can be obtained by a simple rescaling of the distributions of either quark contributing to the interference term. We also present the range of validity of the analytical expression that we have found.

Keywords: Beyond Standard Model, Heavy Quark Physics

ArXiv ePrint: 1311.3977

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP07(2014)142

Web End =10.1007/JHEP07(2014)142

Towards model-independent approach to the analysis of interference e ects in pair production of new heavy quarks

JHEP07(2014)142

Contents

1 Introduction 1

2 Analytical estimation of the interference e ects for pair vector-like quarks production 22.1 Analytical master formula for the interference 22.2 Region of validity of the approximation 4

3 Numerical results 83.1 Total cross section 83.2 Di erential distributions 93.3 Validity range of the model-independent approach and master formula forthe interference 10

4 Conclusions 13

1 Introduction

The discovery of a Higgs boson [1, 2] has essentially excluded a fourth generation of chiral quarks [3, 4], thus shifting the focus of new heavy quark searches towards vector-like quarks (QV s). The latter are heavy spin 1/2 particles that transform as triplets under colour and whose left- and right-handed couplings have the same Quantum Chromo-Dynamics (QCD) and Electro-Weak (EW) quantum numbers. These states are predicted by various theoretical models (composite Higgs models [512], models with extra dimensions, little Higgs models [13, 14], models with gauging of the avour group [1518], non-minimal supersym-metric extensions of the Standard Model (SM) [1924], Grand Unied Theories [25, 26]) and can be observed in a large number of nal states, depending on how they interact with SM particles (see for example [2729, 32, 33] for general reviews).

Usually experimental searches for vector-like quarks adopt a phenomenological approach, assuming that only one new QV state is present beyond the SM and, in order to be as model independent as possible, searches usually consider QCD pair production, although very recently single production has also been explored [35]. Most models, however, predict in general the existence of a new quark sector, which implies the presence of more than one new coloured state, some of which being possibly degenerate or nearly degenerate. If two or more quarks of a given model can decay to the same nal state, interference e ects should be considered in order to correctly evaluate the total cross section and the kinematical distributions of the signal. Current bounds on the masses of new states obtained assuming the presence of only one new particle cannot be easily reinterpreted in more complex scenarios containing more than one new quark, unless interference e ects in the total cross section and kinematical distributions are taken into account.

1

JHEP07(2014)142

We show that this can be done through a simple formula, which enables one to correctly model such interference e ects at both inclusive and exclusive levels. The plan of the paper is as follows. In the next section we describe the procedure while in the following one we present our numerical results. Then, we conclude.

2 Analytical estimation of the interference e ects for pair vector-like quarks production

2.1 Analytical master formula for the interference

We will assume throughout the analysis that the new heavy quarks undergo two-body decays to SM particles and we will not consider chain decays of heavy quarks into other new states, possibly including dark matter candidates. This approach is generally valid for models in which the new quarks interact with the SM ones only through Yukawa couplings. Therefore, the new heavy quarks can decay into either SM gauge bosons or the Higgs boson and ordinary quarks. We will assume that avour changing neutral currents are present and therefore decays such as t Zt and t Ht are allowed, alongside t W +b. This

is consistent with the embedding of new QV s in extensions of the SM. If more than one QV species is present in the model, then there are two ways to obtain a given nal state:

A. QiV quarks have the same charge, so a QiV QiV pair decays into the same nal state, e.g., t1,2t1,2 W +W b

b(W +Zbt);

B. QiV quarks have di erent charges but after decay their pair leads to the same nal state, e.g., bb (tW )(tW +) and X5/3

X5/3 (tW +)(tW ).

We have veried that, while the interference in case B can be safely neglected when the masses of the vector-like quarks are much larger than the masses of the decay products (which is usually the case), because of the largely di erent kinematics of the nal states, case A has to be considered carefully. It is worth mentioning that, for the classes of models under consideration, we have quarks of identical charge and with couplings to the same particles, so that the e ects of the mixing between such quarks at loop level could be important and should (eventually) be taken into account. These e ects are model-dependent though and involve computation of loops that may contain states belonging to new sectors (e.g., new gauge bosons). In this paper we assume that these e ects can be computed and that particle wave-function as well as Feynman rules are already formulated for mass-eigenstates, i.e., the masses and widths that we will be using are those obtained after computing the rotations of the states due to the one-loop mixing terms, so that interference e ects can then be explored in a model-independent way.

The measure of the interference between QiV and QjV pairs of species i and j decaying into the same nal state can be dened by the following simple expression

Fij = intiji + j =

totij (i + j)

i + j =

JHEP07(2014)142

totij

i + j 1 (2.1)

where totij is the total cross section of QiV and QjV pair production including their interference, the i,js are their individual production rates while intij represents the value of the interference.

2

The interference term Fij ranges from 1 to 1. Completely constructive interference

is obviously achieved when intij = i + j, while completely destructive interference is obtained when intij = (i + j).

It is known that, under very general hypotheses, the couplings of QV s with SM quarks are dominantly chiral and that the chirality of the coupling depends on the QV representation under SU(2) [27, 2932]. If the QV belongs to a half-integer representation (doublets, quadruplets, . . . ) couplings are dominantly right-handed while, if the QV belongs to an integer representation (singlets, triplets, . . . ) couplings are mostly left-handed. This feature is valid for a wide range of hypotheses about the mixing between QV s and SM quarks and between QV s themselves. However, if Yukawa couplings between QV s and the Higgs boson are large, it is possible to achieve couplings with non-dominant chiralities.

Our results about the analysis of interference e ects can be applied in both cases, therefore, we divide our study in two parts. Firstly, we show the results for the interference of two ts with the same chiral couplings. Then we generalise the analysis to the case where the couplings of the heavy quarks do not exhibit a dominant chirality.

We would now like to make the ansatz that, in case of chiral new quarks i and assuming small i/mi values, the interference is proportional to the couplings of the new quarks to the nal state particles and to the integral of the scalar part of the propagator. The range of validity of the ansatz in terms of the i/mi ratio is explored in a subsequent section.

If the couplings are chiral for both heavy quarks and the chirality is the same we have

intij 2Re [bracketleftBigg]

gi1gj1gi2gj2

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[parenleftbigg][integraldisplay]

+

dq2PiPj[parenrightbigg]2[bracketrightBigg]

(2.2)

where 1 and 2 refer to the two decay branches (1 corresponding to the quark branch and 2 to the antiquark branch) while the scalar part of the propagator for any new quark i is given by

Pi = 1q2 m2i + imi i. (2.3)

The cross section for pair production of species i only is

i |gi1|2|gi2|2 [parenleftbigg][integraldisplay]

dq2PiPi[parenrightbigg]2

(2.4)

and an analogous expression can be written for species j.

Therefore, the analytical expression which should describe the interference in the case of chiral QV pair production of species i and j followed by their decay into the same nal state, is given by

ij =

2Re

gi1gj1gi2gj2

[parenleftBig][integraltext]

Pi Pj[parenrightBig]2[bracketrightbigg]

. (2.5)

Ultimately, ij should closely describe the true value of the interference term Fij from eq. (2.1) if the ansatz is correct.

3

|gi1|2|gi2|2

[parenleftBigg][integraltext]

Pi Pi

2 + |gj1|2|gj2|2 [parenleftBig][integraltext]

Pj Pj[parenrightBig]2

After integration ij takes the following form:

ij = 8Re[gi1gj1gi2gj2]m2im2j 2i 2j

|gj1|2|gj2|2m2i 2i + |gi1|2|gi2|2m2j 2j

(mi i + mj j)2 [parenleftBig]

m2i m2j[parenrightBig]2

(mi i + mj j)2 +

m2i m2j[parenrightBig]2[parenrightbigg]2

. (2.6)

The previous expression can be generalised when the chirality of the coupling is not predominantly left or right. In the approximation in which the nal states are massless (in practice, neglecting the top mass) only four sub-diagrams give a non-zero contribution, the ones corresponding to considering the following combinations of chiralities: q1, q2, q1, q2=L, L, L, L or L, L, R, R or R, R, L, L or R, R, R, R. If the masses of the nal state objects cannot be neglected, the non-zero combinations would be 16 because any combination of q1 would interfere with any combination of q2, though interferences involving

LR or RL ipping are suppressed by the mass of the quarks in the nal state. Analogously to the previous case, we have numerically proven that neglecting the masses of the nal states is a reasonable assumption in the range of QV masses still allowed by experimental data, hence we will consider the nal state quarks as massless.

The expression in eq. (2.5) can therefore be rewritten in the following way:

abij =

2Re

gai1gaj1gbi2gbj2

[parenleftBig][integraltext]

Pi Pj[parenrightBig]2[bracketrightbigg]

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, ab=LL, LR, RL, RR. (2.7)

After summing over all allowed topologies, we obtain the generalisation of eq. (2.6) as:

genij =

|gai1|2|gbi2|2

[parenleftBigg][integraltext]

Pi Pi

2 +|gaj1|2|gbj2|2 [parenleftBig][integraltext]

Pj Pj[parenrightBig]2

= N abij

Dabij

Pa,b=L,R 2Re

gai1gaj1gbi2gbj2

[parenleftBig][integraltext]

Pi Pj[parenrightBig]2[bracketrightbigg]

=

Pab abijDabij

PabDabij, (2.8)

Pa,b=L,R

|gai1|2|gbi2|2

[parenleftBigg][integraltext]

Pi Pi

2 + |gaj1|2|gbj2|2 [parenleftBig][integraltext]

Pj Pj[parenrightBig]2

which, after integration, becomes

genij =

8Re

[bracketleftBig][parenleftBig]

gLi1gLj1 + gRi1gRj1

[parenrightBig] [parenleftBig]

gLi2gLj2 + gRi2gRj2

[parenrightBig][bracketrightBig]

m2im2j 2i 2j

[parenleftBig][parenleftBig]|

gLj1|2 + |gRj1|2[parenrightBig] [parenleftBig]|

gLj2|2 + |gRj2|2[parenrightBig][parenrightBig]

m2i 2i+

[parenleftBigg][parenleftBigg]|gL

i1

|2 + |gRi1|2

[parenrightbig][parenleftBigg][parenleftBigg]|gL

i2

|2 + |gRi2|2

[parenrightbig][parenrightbig]

m2j 2j

(mi i + mj j)2 [parenleftBig]

m2i m2j[parenrightBig]2

(mi i + mj j)2 +

. (2.9)

2.2 Region of validity of the approximation

When considering the production and decay of di erent heavy quarks which couple to the same SM particles, interference at tree level is not the only one which should potentially be taken into account. Quarks with same quantum numbers can mix at loop level too, which results into the respective mixing matrix of the one-loop corrected propagators and their corresponding interference. Mass and width eigenstates can be obtained by diagonalising

4

m2i m2j[parenrightBig]2[parenrightbigg]2

g

QJ

QJ

JK + JK

IJ + IJ

QK

QI

I, J, K = 1, 2

Figure 1. Pair production of two heavy quarks Q1 and Q2, including loop mixing.

QI AS

mf

BS

QJ

QI AV

mf

BV

QJ

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mS

mV

AS = gSL

I PL +

gSR

I PR

AV = gVL

I PL +

gVR

I PR

J PR

Figure 2. Loop topologies for corrections to quark propagators. The particles in the loop can be any fermion, vector or scalar which are present in the model under consideration.

the respective matrices, but the rotations are in general di erent for these two matrices, therefore mass and width eigenstates may be misaligned. A careful treatment of all such mixing e ects is beyond the scope of this analyis but, in order to be able to apply our results, it is crucial to understand when the mixing e ect can be neglected.

Let us consider the structure of the interference terms for the process of QCD pair production of two heavy quarks, Q1 and Q2, including the one-loop corrections to the quark propagators. From now on we will consider only the imaginary part of the quark self-energies, that give the corrections to the quark widths, and we will assume real couplings for simplicity. A more detailed treatment of mixing e ects under general assumptions in heavy quark pair production will be performed in a dedicated analysis [34]. Considering only the case of s-channel exchange of the gluon for simplicity, and still not including the decays of the heavy quarks, the amplitude of the process depicted in gure 1 is:

M=I( IJ + IJ)P +JV P J( JK + JK)vKMP with I, J, K = 1, 2 (2.10)

where the QCD amplitude terms and colour structure have been factorised into the vertex V and the term MP , the propagators of the quark and antiquarks are P + and P ,

respectively, and represents the loop insertions. The loop contributions depend on the particle content of the model and therefore cannot be evaluated in a model independent way. However, it is straightforward to determine the structure of the loops by noticing that the only allowed topologies are fermion-scalar (fS) and fermion-vector (fV), see gure 2.

These topologies can be evaluated for general masses and couplings of the particles in the loops, and therefore the most general structure of the loop insertion is:

IJ =

XfS loops fSIJ +

XfV loops fVIJ (2.11)

BS = gSL

J PL +

gSR

J PR

BV = gVL

J PL +

gVR

5

where, in Feynman gauge and adopting the Passarino-Veltman functions B0 and B1:

fSIJ =

[parenleftBig][parenleftBigg]

gSL

I

gSL

J mfB0 p2, m2f, m2S

+

gSR

I

gSL

J / pB1

p2, m2f, m2S

[parenrightbig][parenrightBig]

PL+LR, (2.12)

L +LR. (2.13)

When I = J, the loop contributions correspond to a correction to the diagonal quark propagators while, when I 6= J, the loops correspond to the o -diagonal mixing between

the quarks. Without loosing generality, let us consider the I, K = 1, 2 case, for which we can dene two amplitude matrices, corresponding to production of the quarks J = 1 and J = 2 that, through the loop-corrected propagators, become quarks I, K = 1, 2.

The amplitude matrices are:

MJ=1 =1(1 + 11)P +1V P 1(1+ 11)v1MP1(1+ 11)P +1V P 1 12v2MP

2 21P +1V P 1(1+ 11)v1MP2 21P +1V P 1 12v2MP [parenrightBigg]

, (2.14)

I

gVL

fVIJ =

4 gVR

J mfB0

p2, m2f, m2V

2

gVL

I

gVL

J /pB1 p2, m2f, m2V

[parenrightbig][parenrightBig]P

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MJ=2 =1 12P +2V P 2 21v1MP1 12P +2V P 2(1+ 22)v2MP

2(1+ 22)P +2V P 2 21v1MP2(1+ 22)P +2V P 2(1+ 22)v2MP [parenrightBigg]

. (2.15)

The interference contribution of the cross-section can be obtained by contracting elements of one matrix with elements of the other matrix. Some interesting consequences can be derived from the structure of these matrices.

1. It is possible to construct four interference terms by contracting elements with same indices (e.g. MJ=1|(1,1) with MJ=2|(1,1)) due to the fact that the quarks in the nal

state are the same. At lowest order these interference terms will always contain two o -diagonal loop corrections.

2. Any element of one matrix can be contracted with any element of the other matrix only when considering also the decays of the quarks, there xing specic decay channels for the quark and antiquark branches. This way it is possible to obtain 16 interference combinations. The order of the interference term and the number of o -diagonal mixing contributions, however, will not always be the same, depending on the contraction. In particular, when contracting the element (1,1) of the MJ=1 matrix with the element (2,2) of the MJ=2 matrix, there are no o -diagonal loop

mixings involved and the contraction after the quark decays will be given by a pure tree level contribution plus diagonal loop corrections while, when contracting the element (2,2) of the MJ=1 matrix with the element (1,1) of the MJ=2 matrix, there

are 4 o -diagonal loop mixings involved, so that this process, which has mixing terms to a higher power, is expected to be suppressed.

It is interesting to notice that, in the case of same-element contractions before quark decays (case 1), the order of the process is the same as in the case of contractions after quark decays of the element (1,1) of the MJ=1 matrix with the element (2,2) of the MJ=2 matrix

(case 2). Therefore, the 4 interference contributions of case 1 can be competitive with the tree-level interference term after quark decay. However, if the o -diagonal contributions to

6

the mixing matrix are negligible with respect to the diagonal elements, the two amplitude matrices reduce to:

!, (2.16)

. (2.17)

In this case the same-element contraction of case 1 do not enter the determination of the interference terms and the lowest order contribution is given by contracting the only nonzero elements of the matrices at tree level after the decays of the quarks. In other words, the analytical description of the interference developed in the previous section can only be applied in the case of suppressed or negligible mixing between the heavy quarks. One should note that the requirement of suppression of o -diagonal mixing can be potentially quite restrictive, since it will take place in case of cancellation of loop contributions in the kinematic p2 M2Q region where the couplings of the heavy quarks are chosen to

compensate the di erent values of the loop integrals. The verication of such a case is eventually model-dependent and requires computing the mixing matrix structure, which in turn depends on the particle content of the model. For example in case of the o -diagonal contributions to the propagators of two top partners T1 and T2 that only couple to the third family of SM quarks and with all SM gauge bosons and the Higgs boson, and requiring their sum to be suppressed with respect to the sum of the diagonal contributions, we obtain the following relation:

IJ = tHIJ + tZIJ + bWIJ + tG0IJ + bG+IJ { II, JJ} (2.18)

with I, J = 1, 2 and I 6= J. The suppression of the o -diagonal contribution depends

on all the masses and couplings involved, plus it also depends on the p2 of the external heavy quarks. However, if it is possible to nd coupling congurations which satisfy the relation for a large p2 region, our approach can be safely adopted. A detailed numerical treatment of this relation for di erent particle contents and coupling values is beyond the scope of this preliminary analysis, but it will be developed in a future one [34]. It is also interesting to notice that, if the mass and width eigenvalues are not misaligned, it is possible to diagonalise the matrix of the propagators and dene new states with denite mass and eigenstates. In this case it is possible to consider the exact amplitude matrix,

MJ=1 =1

P +1V P 1v1

, (2.20)

then compute the tree-level interference after the decays of the quarks with the method developed in the previous section, but considering quarks with loop-corrected masses and widths. Again, this is a specic situation, but it is a further case when the relations stuided in this paper can be applied.

7

MJ=1

1(1 + 11)P +1V P 1(1 + 11)v1MP 0

0 0

0 002(1 + 22)P +2V P 2(1 + 22)v2MP [parenrightBigg]

MJ=2

JHEP07(2014)142

!, (2.19)

MJ=2 = 0 0

02 P +2V P 2v2

MP [parenrightBigg]

b

W +

Z

t

p

p

Figure 3. Pair production of a pair of t QV s and subsequent decay into a bW +tZ nal state.

3 Numerical results

3.1 Total cross section

We rst consider the production and decay rates of two ts pairs decaying into W +b and Zt, see gure 3, i.e., we consider the 2 4 process

pp titi W +bZt, i = 1, 2, (3.1)

with the chirality of the couplings being the same for the two states. This process has been chosen to provide a concrete example; in general, vector-like quarks can also decay into the Higgs boson, but we have xed a specic nal state to perform the simulations. Selecting di erent nal states involving decays into Higgs would give analogous results.

We have performed a scan on the QV s couplings for di erent values of masses and splitting between the two ts and we have obtained the value of the interference term (2.1) through numerical simulation with MadGraph5 [36] and alternatively cross-checked via CalcHEP3.4 [37]. The results are shown in gure 4 (left frame), where it is possible to notice a remarkable linear correlation between Fij and the expression in eq. (2.6).

If the chirality of the couplings of t1 and t2 with respect to the SM quarks is opposite, interference e ects can arise when the masses of the quarks in the nal state are not negligible, as is in the case of decay to top quarks. Considering a scenario where t1s decays predominantly to ZtL and t2 does so in ZtR, then the interference between tL and tR may in principle become relevant. We have numerically veried, however, that in case the chirality of the two QV is opposite, the interference e ect between massive nal states is always negligible, unless the QV s masses approach the threshold of the nal state. This case implies, however, very light QV s, with masses of the order of 300 GeV, and this range is already excluded by experimental searches.

We show in gure 4 (right frame) the results for the analogous process (3.1) where both chiralities are now present in the couplings of QV s: this process is described by the generalised eq. (2.9). Interference e ects between nal state quarks of di erent chiralities become relevant when the masses of the heavy quarks are close to the top mass, but, as already stressed, this scenario has been tested only to show the appearance of chirality ipping interference e ects, since such a low value for the mass of the heavy quarks is already experimentally excluded.

8

t

t

JHEP07(2014)142

Chiral couplings General couplings

ppW+b Zt mt =300 GeV NWF=0.01

ppW+b Zt mt =300 GeV NWF=0.01

1.0

1.0

0.5

0.5

F 12

0.0

F 12

0.0

-0.5

-0.5

- -1.0 -0.5 0.0 0.5 1.0

1.0

- -1.0 -0.5 0.0 0.5 1.0

1.0

JHEP07(2014)142

12

12

ppW+b Zt mt =600 GeV NWF=0.01

ppW+b Zt mt =600 GeV NWF=0.01

1.0

1.0

0.5

0.5

F 12

0.0

F 12

0.0

-0.5

-0.5

- -1.0 -0.5 0.0 0.5 1.0

1.0

- -1.0 -0.5 0.0 0.5 1.0

1.0

12

12

ppW+b Zt mt =1000 GeV NWF=0.01

ppW+b Zt mt =1000 GeV NWF=0.01

1.0

1.0

0.5

0.5

F 12

0.0

F 12

0.0

-0.5

-0.5

- -1.0 -0.5 0.0 0.5 1.0

1.0

- -1.0 -0.5 0.0 0.5 1.0

1.0

12

12

Figure 4. Interference term Fij as a function of ij. In the left frame the couplings are chiral while in the right one they are general. The cyan-dashed line is the bisector in the ij Fij plane.

Blue points are the results of the scan on the couplings for mt1 = 300, 600, 1000 GeV, with di erent values of the mass splitting between t1 and t2. The Narrow Width Factor (NWF) is the upper limit on max( t1 /mt

1 , t

2 / mt

2 ) for each point of the scan.

3.2 Di erential distributions

The results of the previous sections only apply to the total cross section of the process of pair production and decay of the heavy quarks. However, it is necessary to evaluate how kinematic distributions are a ected by the presence of interference terms, as experimental e ciencies of a given search may be largely di erent if the kinematics of the nal state is

9

not similar to the case without interference. To evaluate the contribution of interference we have considered the process pp W +bZt, with subsequent semileptonic decay of the

top, mediated by two heavy top-like partners t1 and t2 in three limiting cases:

degenerate masses (mt

handed);

1,2 = 600 GeV) and couplings with opposite chirality;

non-degenerate masses (mt

with same chirality (both left-handed).

The results are shown in gure 5, where we display the HT (scalar sum of the transverse momenta of jets) and

ET (missing transverse energy) di erential distributions. When the interference is maximal, all distributions have exactly the same features, that is, the distributions including interference can be obtained by a rescaling of the distributions for production of the two heavy quarks using (1 + ij) for the rescaling factor: this relation comes from considering eq. (2.1) and the linear correlation between Fij and ij veried in the previous section. Therefore our results for the total cross section can also be applied at di erential level and, specically, it is possible to apply the same experimental e ciencies to the case of a single heavy quark or to the case with degenerate quarks with couplings of identical chirality. In contrast, in the two other scenarios we have considered, where interference is negligible, the distributions for production of either t1 or t2 exhibit di erent features and the distribution of the total process is, for each bin, simply the sum of the distributions of the two heavy quarks (i.e. the rescaling factor is 1 because kij 0).

Same patterns are seen for all other di erential distributions that we have investigated: (pseudo)rapidity, cone separation, etc.

As a nal remark, we may ask how much the range of the possible values for the interference term drops by increasing the mass splitting between the heavy quarks and, therefore, when should we consider the interference as always negligible. In gure 6 it is possible to notice that the range of values for the parameter 12 drops extremely fast with the mass splitting and depends on the value of the NWF. The range of the interference contributions, however, becomes smaller than 10% in a region of mass splitting where the shapes of the distributions can be safely considered as equivalent.

3.3 Validity range of the model-independent approach and master formula for the interference

In this subsection we discuss the range of validity of the analytical formula for ij describing the interference e ect. Our ansatz was made under the assumption of small /m ratios, which, in terms of probability (e.g. amplitude square), means that the QCD production part of the QV s and their subsequent decay can be factorised. We then took advantage of this consideration by making this factorisation already at amplitude level and writing therefore the interference, eq. (2.2), and pair production, eq. (2.4), contribution to the total cross section as a modulus squared of quantities that do not involve the QCD production

10

1,2 = 600 GeV) and couplings with same chirality (both left-

degenerate masses (mt

1 = 600 GeV, mt

2 = 1.1mt

1 = 660 GeV) and couplings

JHEP07(2014)142

Scalar sum of transverse momentum Missing transverse energy

Degenerate masses, same chirality: F

contribution of t contribution of tTotal with interference

Degenerate masses, same chirality: F

=0.999

12

=0.999

12

Degenerate masses, same chirality: F

contribution of t contribution of tTotal with interference

Degenerate masses, same chirality: F

=0.999

12

=0.999

12

[pb/GeV] /dH s d

/dMET [pb/GeV] s d

0.8

10

0.7

0.6

10

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

JHEP07(2014)142

0 0 500 1000 1500 2000 2500 3000

0 0 50 100 150 200 250 300 350 400 450 500

H

[GeV]

MET [GeV]

Degenerate masses, opposite chirality: F

contribution of t contribution of tTotal with interference

Degenerate masses, opposite chirality: F

=0.0008

12

=0.0008

12

Degenerate masses, opposite chirality: F

contribution of t contribution of tTotal with interference

Degenerate masses, opposite chirality: F

=0.0008

12

=0.0008

12

0.4

10

[pb/GeV] /dH s d

0.35

/dMET [pb/GeV] s d

0.35

10

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 0 500 1000 1500 2000 2500 3000

0 0 50 100 150 200 250 300 350 400 450 500

H

[GeV]

MET [GeV]

m

m

/m

/m

=1.1, same chirality: F

=1.1, same chirality: F

=-0.0003

12

=-0.0003

12

m

m

/m

/m

=1.1, same chirality: F

=1.1, same chirality: F

=-0.0003

12

=-0.0003

12

t

t

t

t

t

t

t

t

0.3 10

0.25

10

[pb/GeV] /dH s d

contribution of t contribution of tTotal with interference

0.25

/dMET [pb/GeV] s d

contribution of t contribution of tTotal with interference

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 0 500 1000 1500 2000 2500 3000

0 0 50 100 150 200 250 300 350 400 450 500

H

[GeV]

MET [GeV]

Figure 5. Di erential distributions for HT and

ET for the process pp W +bZt W +bZ

be

e

in three di erent scenarios: degenerate masses and couplings with same chirality (top); degenerate masses and couplings with opposite chirality (middle); non-degenerate masses (mt2 = 1.1mt

1 ) and

couplings with same chirality (bottom). Here, mt1 has been xed to 600 GeV. The values of the interference term F12 are shown for each scenario.

part, then using then these two relations to dene our ij parameter in eq. (2.5). This concept of factorisation is valid just in the limit /m 0, for which, however, there

will be no decay of the QV s and therefore no interference at all. It is nonetheless clear that this approximation of factorisation of production and decay will be the more accurate

11

JHEP07(2014)142

Figure 6. The range of the interference contributions with respect to the mass splitting between the heavy quarks for di erent values of the NWF. Notice the di erent scales of the x axis.

ppW+ bZ

t mt =1000 GeV mt =1001 GeV

ppW+ bZ

t mt =1000 GeV mt =1001 GeV

1.0

NWF

0.01 > >0. 0.03 > >0.01 0.10 > >0.03 0.30 > >0.10

NWF

0.01 > >0. 0.03 > >0.01 0.10 > >0.03 0.30 > >0.10

1.4

0.5

1.2

F ij

0.0

1.0

(+)(1+)

0.8

0.5

0.6

1.0

1.0 0.5 0.0 0.5 1.0 ij

1.0 0.5 0.0 0.5 1.0 ij

Figure 7. Fij versus ij (left) and tot

(1+2)(1+ ij) versus ij (right) for various values of the NWF for the pp W +bZt process.

the more this ratio is closer to zero. In fact, in the previous subsections we have shown that the formula for ij reproduces the true interference Fij very accurately in the case of

NWF= /m = 0.01. It is however very informative to explore the range of validity of our ansatz in function of the NWF parameter, especially in view of practical applications of our method.

In gure 7 (left) we present results for Fij versus ij for values of the NWF in the0.00.3 range for the pp W +bZt process. One can see that our description of the

interference remains at a quite accurate level for NWF below about 10% while already in the range 10%30% one can see non-negligible deviations from the analytic formula predictions, i.e., ij, as compared to the true value of the interference, Fij. The triangle

shape of the pattern of the left frame of gure 7 is simply related to the fact that, in case of large negative interference, the totij value is close to zero. Therefore, even in case of large relative deviations, the predicted value of totij will be still close to zero, forcing Fij to be around 1, according to eq. (2.1), even in case of large values of the

12

NWF parameter. Therefore, it is important to look at the complementary plot presenting

tot(1+2)(1+ ij) versus ij shown in gure 7 (right). One can see that deviations of the cross-section predicted by the master formula, (1 + 2)(1 + ij), from the real one, tot, depends only on the value of NWF. For large values of NWF one can also see that tot is

below (1 + 2)(1 + ij), which is related to the fact that in case of tot the pure Breit-Wigner shape of the ti resonances is actually distorted and suppressed on the upper end due to steeply falling parton distribution functions. Furthermore, one should note that the quite accurate description of the interference found at the integrated level for NWF < 0.1 remains true at di erential level too. Finally, we remark that the multi-parametric scan was done using CalcHEP3.4 on the HEPMDB database [38], where the model studied here can be found under the http://hepmdb.soton.ac.uk/hepmdb:1113.0149

Web End =http://hepmdb.soton.ac.uk/hepmdb:1113.0149 link.

4 Conclusions

We have studied the role of interference in the process of pair production of new heavy (vector-like) quarks. Considering such interference e ects is crucial for the reinterpretation of the results of experimental searches of new quarks decaying to the same nal state in the context of models with a new quark sector, which is usually not limited to the presence of only one heavy quark. We have shown that, if the small /m approximation holds, and therefore it is possible to factorise the production and decay of the new quarks, the interference contribution can be described by considering a parameter which contains only the relevant couplings and the scalar part of the propagators of the new quarks.

We have obtained a remarkably accurate description of the exact interference (described by the term F12 dened in eq. (2.1)) using a simple analytical formula for the parameter ij dened in eq. (2.6). This description holds regardless of the chiralities of the couplings between the new and SM quarks, eq. (2.9). This means that it is possible to analytically estimate, with very good accuracy, the interference contribution to the pair production of two (and possibly more) quarks pairs decaying into the same nal state, once couplings, total widths and masses are known, without performing a dedicated simulation or a full analytical computation. We have also discussed the region of validity of this approximation in connection to the mixing e ects at the loop-level contribution to a heavy quark self-energy which could potentially lead to a non-negligible interference. Therefore, in order to use the analytical formula for the interference we have derived, one should verify that the o -diagonal contributions to the propagators are suppressed and check that the relation analogous to eq. (2.18) takes place for the particular model under study.

We have veried that also at the level of di erential distributions it is possible to obtain the distributions including interference by a simple rescaling of those of the heavy quarks decaying to the given nal state. Finally, we have checked that the linear correlation does not hold anymore for large values of the /m ratio, while it has been veried that for a NWF less than 10% (which is very typical for all classes of models with QV s), the expressions for ij do indeed provide an accurate description of the interference term. When interference e ects are relevant and in the range of validity of our expressions, it is therefore possible to apply the same experimental e ciencies used for individual quark pairs to the full process of production and decay of two pairs of new quarks.

13

JHEP07(2014)142

Acknowledgments

The authors would like to thank M. Buchkremer, G. Cacciapaglia, A. Deandrea andS. De Curtis for useful discussions. They also thank A. Pukhov for a quick response which helped to improve the interference treatment within CalcHEP. DB, AB, SM and LP are nanced in part through the NExT Institute. DB and LP would like to thank the Galileo Galilei Institute (GGI) in Florence for hospitality while part of this work was carried out. JB thanks the University of Southampton for the Summer Student programme support. AB and JB acknowledge the use of the HEPMDB and IRIDIS HPC Facility at the University of Southampton in the completion of this study.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, http://dx.doi.org/10.1016/j.physletb.2012.08.020

Web End =Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.7214

Web End =INSPIRE ].

[2] CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, http://dx.doi.org/10.1016/j.physletb.2012.08.021

Web End =Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.7235

Web End =INSPIRE ].

[3] A. Djouadi and A. Lenz, Sealing the fate of a fourth generation of fermions, http://dx.doi.org/10.1016/j.physletb.2012.07.060

Web End =Phys. Lett. B 715 (2012) 310 [arXiv:1204.1252] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1204.1252

Web End =INSPIRE ].

[4] O. Eberhardt et al., Impact of a Higgs boson at a mass of 126 GeV on the standard model with three and four fermion generations, http://dx.doi.org/10.1103/PhysRevLett.109.241802

Web End =Phys. Rev. Lett. 109 (2012) 241802 [arXiv:1209.1101] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.1101

Web End =INSPIRE ].

[5] B.A. Dobrescu and C.T. Hill, Electroweak symmetry breaking via top condensation seesaw, http://dx.doi.org/10.1103/PhysRevLett.81.2634

Web End =Phys. Rev. Lett. 81 (1998) 2634 [http://arxiv.org/abs/hep-ph/9712319

Web End =hep-ph/9712319 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9712319

Web End =INSPIRE ].

[6] R.S. Chivukula, B.A. Dobrescu, H. Georgi and C.T. Hill, Top quark seesaw theory of electroweak symmetry breaking, http://dx.doi.org/10.1103/PhysRevD.59.075003

Web End =Phys. Rev. D 59 (1999) 075003 [http://arxiv.org/abs/hep-ph/9809470

Web End =hep-ph/9809470 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9809470

Web End =INSPIRE ].

[7] H.-J. He, C.T. Hill and T.M.P. Tait, Top quark seesaw, vacuum structure and electroweak precision constraints, http://dx.doi.org/10.1103/PhysRevD.65.055006

Web End =Phys. Rev. D 65 (2002) 055006 [http://arxiv.org/abs/hep-ph/0108041

Web End =hep-ph/0108041 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0108041

Web End =INSPIRE ].

[8] C.T. Hill and E.H. Simmons, Strong dynamics and electroweak symmetry breaking,http://dx.doi.org/10.1016/S0370-1573(03)00140-6

Web End =Phys. Rept. 381 (2003) 235 [Erratum ibid. 390 (2004) 553-554] [http://arxiv.org/abs/hep-ph/0203079

Web End =hep-ph/0203079 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0203079

Web End =INSPIRE ].

[9] K. Agashe, R. Contino and A. Pomarol, The minimal composite Higgs model, http://dx.doi.org/10.1016/j.nuclphysb.2005.04.035

Web End =Nucl. Phys. B 719 (2005) 165 [http://arxiv.org/abs/hep-ph/0412089

Web End =hep-ph/0412089 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0412089

Web End =INSPIRE ].

[10] R. Contino, L. Da Rold and A. Pomarol, Light custodians in natural composite Higgs models, http://dx.doi.org/10.1103/PhysRevD.75.055014

Web End =Phys. Rev. D 75 (2007) 055014 [http://arxiv.org/abs/hep-ph/0612048

Web End =hep-ph/0612048 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0612048

Web End =INSPIRE ].

[11] R. Barbieri, B. Bellazzini, V.S. Rychkov and A. Varagnolo, The Higgs boson from an extended symmetry, http://dx.doi.org/10.1103/PhysRevD.76.115008

Web End =Phys. Rev. D 76 (2007) 115008 [arXiv:0706.0432] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0706.0432

Web End =INSPIRE ].

[12] C. Anastasiou, E. Furlan and J. Santiago, Realistic Composite Higgs Models, http://dx.doi.org/10.1103/PhysRevD.79.075003

Web End =Phys. Rev. D 79 (2009) 075003 [arXiv:0901.2117] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0901.2117

Web End =INSPIRE ].

14

JHEP07(2014)142

[13] N. Arkani-Hamed, A.G. Cohen, E. Katz and A.E. Nelson, The Littlest Higgs, http://dx.doi.org/10.1088/1126-6708/2002/07/034

Web End =JHEP 07 (2002) 034 [http://arxiv.org/abs/hep-ph/0206021

Web End =hep-ph/0206021 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0206021

Web End =INSPIRE ].

[14] M. Schmaltz and D. Tucker-Smith, Little Higgs review,http://dx.doi.org/10.1146/annurev.nucl.55.090704.151502

Web End =Ann. Rev. Nucl. Part. Sci. 55 (2005) 229 [http://arxiv.org/abs/hep-ph/0502182

Web End =hep-ph/0502182 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0502182

Web End =INSPIRE ].

[15] A. Davidson and K.C. Wali, Family Mass Hierarchy From Universal Seesaw Mechanism, http://dx.doi.org/10.1103/PhysRevLett.60.1813

Web End =Phys. Rev. Lett. 60 (1988) 1813 [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,60,1813

Web End =INSPIRE ].

[16] K.S. Babu and R.N. Mohapatra, A solution to the strong CP problem without an axion, http://dx.doi.org/10.1103/PhysRevD.41.1286

Web End =Phys. Rev. D 41 (1990) 1286 [http://inspirehep.net/search?p=find+J+Phys.Rev.,D41,1286

Web End =INSPIRE ].

[17] B. Grinstein, M. Redi and G. Villadoro, Low Scale Flavor Gauge Symmetries, http://dx.doi.org/10.1007/JHEP11(2010)067

Web End =JHEP 11 (2010) 067 [arXiv:1009.2049] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.2049

Web End =INSPIRE ].

[18] D. Guadagnoli, R.N. Mohapatra and I. Sung, Gauged Flavor Group with Left-Right Symmetry, http://dx.doi.org/10.1007/JHEP04(2011)093

Web End =JHEP 04 (2011) 093 [arXiv:1103.4170] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.4170

Web End =INSPIRE ].

[19] T. Moroi and Y. Okada, Radiative corrections to Higgs masses in the supersymmetric model with an extra family and antifamily, http://dx.doi.org/10.1142/S0217732392000124

Web End =Mod. Phys. Lett. A 7 (1992) 187 [http://inspirehep.net/search?p=find+J+Mod.Phys.Lett.,A7,187

Web End =INSPIRE ].

[20] T. Moroi and Y. Okada, Upper bound of the lightest neutral Higgs mass in extended supersymmetric Standard Models, http://dx.doi.org/10.1016/0370-2693(92)90091-H

Web End =Phys. Lett. B 295 (1992) 73 [http://inspirehep.net/search?p=find+J+Phys.Lett.,B295,73

Web End =INSPIRE ].

[21] K.S. Babu, I. Gogoladze, M.U. Rehman and Q. Sha, Higgs Boson Mass, Sparticle Spectrum and Little Hierarchy Problem in Extended MSSM, http://dx.doi.org/10.1103/PhysRevD.78.055017

Web End =Phys. Rev. D 78 (2008) 055017 [arXiv:0807.3055] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.3055

Web End =INSPIRE ].

[22] S.P. Martin, Extra vector-like matter and the lightest Higgs scalar boson mass in low-energy supersymmetry, http://dx.doi.org/10.1103/PhysRevD.81.035004

Web End =Phys. Rev. D 81 (2010) 035004 [arXiv:0910.2732] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0910.2732

Web End =INSPIRE ].

[23] P.W. Graham, A. Ismail, S. Rajendran and P. Saraswat, A little solution to the little hierarchy problem: a vector-like generation, http://dx.doi.org/10.1103/PhysRevD.81.055016

Web End =Phys. Rev. D 81 (2010) 055016 [arXiv:0910.3020] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0910.3020

Web End =INSPIRE ].

[24] S.P. Martin, Raising the Higgs mass with Yukawa couplings for isotriplets in vector-like extensions of minimal supersymmetry, http://dx.doi.org/10.1103/PhysRevD.82.055019

Web End =Phys. Rev. D 82 (2010) 055019 [arXiv:1006.4186] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.4186

Web End =INSPIRE ].

[25] J.L. Rosner, E6 and Exotic Fermions, Comments Nucl. Part. Phys. 15 (1986) 195 [http://inspirehep.net/search?p=find+IRN+1355503

Web End =INSPIRE ].

[26] R.W. Robinett, On the Mixing and Production of Exotic Fermions in E6, http://dx.doi.org/10.1103/PhysRevD.33.1908

Web End =Phys. Rev. D 33 (1986) 1908 [http://inspirehep.net/search?p=find+J+Phys.Rev.,D33,1908

Web End =INSPIRE ].

[27] J.A. Aguilar-Saavedra, Identifying top partners at LHC, http://dx.doi.org/10.1088/1126-6708/2009/11/030

Web End =JHEP 11 (2009) 030 [arXiv:0907.3155] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0907.3155

Web End =INSPIRE ].

[28] Y. Okada and L. Panizzi, LHC signatures of vector-like quarks,

http://dx.doi.org/10.1155/2013/364936

Web End =Adv. High Energy Phys. 2013 (2013) 364936 [arXiv:1207.5607] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.5607

Web End =INSPIRE ].

[29] A. De Simone, O. Matsedonskyi, R. Rattazzi and A. Wulzer, A First Top Partner Hunters Guide, http://dx.doi.org/10.1007/JHEP04(2013)004

Web End =JHEP 04 (2013) 004 [arXiv:1211.5663] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.5663

Web End =INSPIRE ].

[30] F. del Aguila and M.J. Bowick, The Possibility of New Fermions With I = 0 Mass, http://dx.doi.org/10.1016/0550-3213(83)90316-4

Web End =Nucl. Phys. B 224 (1983) 107 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B224,107

Web End =INSPIRE ].

[31] F. del Aguila, M. Prez-Victoria and J. Santiago, Observable contributions of new exotic quarks to quark mixing, http://dx.doi.org/10.1088/1126-6708/2000/09/011

Web End =JHEP 09 (2000) 011 [http://arxiv.org/abs/hep-ph/0007316

Web End =hep-ph/0007316 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0007316

Web End =INSPIRE ].

15

JHEP07(2014)142

[32] M. Buchkremer, G. Cacciapaglia, A. Deandrea and L. Panizzi, Model Independent Framework for Searches of Top Partners, http://dx.doi.org/10.1016/j.nuclphysb.2013.08.010

Web End =Nucl. Phys. B 876 (2013) 376 [arXiv:1305.4172] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.4172

Web End =INSPIRE ].

[33] J.A. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer and M. Prez-Victoria, Handbook of vectorlike quarks: Mixing and single production, http://dx.doi.org/10.1103/PhysRevD.88.094010

Web End =Phys. Rev. D 88 (2013) 094010 [arXiv:1306.0572] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.0572

Web End =INSPIRE ].

[34] D. Barducci, A. Belyaev, M. Buchkremer, S. Moretti, L. Panizzi and H. Prager, Interference e ects in QCD pair production of heavy quarks at the LHC: the role of loop mixing, work in progress.

[35] ATLAS collaboration, Search for single b-quark production with the ATLAS detector at s = 7 TeV, http://dx.doi.org/10.1016/j.physletb.2013.03.016

Web End =Phys. Lett. B 721 (2013) 171 [arXiv:1301.1583] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1301.1583

Web End =INSPIRE ].

[36] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, MadGraph 5: Going Beyond, http://dx.doi.org/10.1007/JHEP06(2011)128

Web End =JHEP 06 (2011) 128 [arXiv:1106.0522] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.0522

Web End =INSPIRE ].

[37] A. Belyaev, N.D. Christensen and A. Pukhov, CalcHEP 3.4 for collider physics within and beyond the Standard Model, http://dx.doi.org/10.1016/j.cpc.2013.01.014

Web End =Comput. Phys. Commun. 184 (2013) 1729 [arXiv:1207.6082] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.6082

Web End =INSPIRE ].

[38] G. Brooijmans et al., Les Houches 2011: Physics at TeV Colliders New Physics Working Group Report, arXiv:1203.1488 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.1488

Web End =INSPIRE ].

JHEP07(2014)142

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