# Tevatron Task Force

## 

## 

## 

## 

## Talk outline

1.  Luminosity lifetime

    a.  Current scenario

    b.  Final Run II scenario

2.  Beam instabilities

    a.  Transverse head-tail

    b.  "Dancing bunches"

3.  Beam-beam effects

4.  Particle loss at injection

5.  The source of background in the particle physics detectors

> Conclusions

## Luminosity Lifetime 

The model takes into account the major beam heating and particle loss
mechanisms

-   **[Phenomena taken into account]{.underline}**

    -   Interaction with residual gas

        -   Emittance growth due to electromagnetic scattering

        -   Particle loss due to nuclear and electromagnetic interaction

    -   Particle interaction in IPs (proportional to the luminosity)

        -   Emittance growth due to electromagnetic scattering

        -   Particle loss due to nuclear and electromagnetic interaction

    -   IBS

        -   Energy spread growth and emittance growth due to multiple
            scattering

    -   Bunch lengthening due to RF noise

    -   Particle loss from the bucket due to heating of longitudinal
        degree of freedom

-   **[Phenomena ignored in the model]{.underline}**

    -   Beam-beam effects

    -   Non-linearity of the lattice

    -   Diffusion amplification by coherent effects

-   Thus, it can be considered as **the best-case scenario**

    -   It describes well our best present stores

### Interaction with Residual Gas (Luminosity lifetime backup slide)

**[Beam lifetime]{.underline}**

$$\tau_{\text{scat}^{- 1}} = \frac{2\pi\text{cr}_{p^{2}}}{\gamma^{2}\beta^{3}}\left( \sum_{i}^{}{n_{i}Z_{i}\left( Z_{i} + 1 \right)} \right)\left( \frac{\overline{\beta_{x}}}{\varepsilon_{\text{mx}}} + \frac{\overline{\beta_{y}}}{\varepsilon_{\text{my}}} \right) + \sum_{i}^{}{n_{i}\sigma_{i}c\beta}$$

> where
> $\overline{\beta_{x,y}} = \frac{1}{C}\int_{}^{}{\beta_{x,y}\text{ds}}\  \approx \ \text{70}\ m$
>
> *~mx~*, *~my~* -- acceptances are chosen to be 6^2^20 mm mrad

-   Average vacuum has been adjusted to match the beam lifetime and the
    emittance growth rate for small intensity beam, P=110^-9^ Torr of
    N~2~ equivalent

    -   Coulomb scattering (\~15350 hour)

    -   Nuclear absorption (\~306 hour)

    -   Total gas scattering lifetime (\~300 hour)

**[Emittance growth time]{.underline}**

$$\frac{{d\varepsilon}_{x,y}}{\text{dt}} = \frac{2\pi\text{cr}_{p^{2}}}{\gamma^{2}\beta^{3}}\left( \sum_{i}^{}{n_{i}Z_{i}\left( Z_{i} + 1 \right)L_{C_{i}}} \right)\overline{\beta_{x,y}}$$

$$\frac{{d\varepsilon}_{x}}{\text{dt}} \approx \frac{{d\varepsilon}_{y}}{\text{dt}} \approx 0\text{.}\text{169}\ \text{mm}\ \text{mrad/hour}$$

### Longitudinal Diffusion (Luminosity lifetime backup slide)

### Diffusion equation, $\frac{\partial t}{\partial t} = \frac{D}{2}\left( \frac{\partial f}{\partial E} + I\frac{\partial E}{\partial I}\frac{\partial^{2}f}{\partial E^{2}} \right)$ , describing the distribution function transformation under action of constant diffusion, $D(I) = D,$ is solved numerically. 

### ![](media/image1.wmf)![](media/image2.wmf)

*f*(*I*(*E*)) *f*(*I*) *f*() *f~p~* (*p*)

Distribution functions as functions of the beam energy, action,
longitudinal coordinate and the particle momentum deviation

where: $\int_{0}^{I_{\text{max}}}{f(I)\text{dI}} = 1$ ,
$f_{\varphi}(\varphi) = \int_{- p_{\text{max}}(\varphi)}^{p_{\text{max}}(\varphi)}{f(I(\varphi,p))\text{dp}}$
,
$f_{p}(p) = \int_{- \varphi_{\text{max}}(p)}^{\varphi_{\text{max}}(p)}{f(I(\varphi,p))d\varphi}$
**.**

### Longitudinal Diffusion -- continue (Luminosity lifetime backup slide)

![](media/image3.wmf)![](media/image4.wmf)

### This numerical solution yields approximate relationships between the bunch parameters:

$$\sigma_{s} \approx \Gamma_{s}\sigma_{\frac{\Delta p}{p}}\left( 1 + \frac{1}{4}\left( \frac{2\sigma_{\frac{\Delta p}{p}}}{\frac{\Delta P}{P}|_{\text{sep}}} \right)^{2} + \frac{1}{6}\left( \frac{2\sigma_{\frac{\Delta p}{p}}}{\frac{\Delta P}{P}|_{\text{sep}}} \right)^{3} \right)$$

$$\frac{1}{N}\frac{\text{dN}}{\text{dt}} \approx \frac{2\text{.}\text{425}\ \left( 2\text{πσ}_{s} \right)^{7}}{\lambda_{\text{RF}^{7}} + 1\text{.}\text{65}\ \left( 2\text{πσ}_{s} \right)^{7}}\left( \left( \frac{2{\pi\Gamma}_{s}}{\lambda_{\text{RF}}} \right)^{2}\frac{d\left( \sigma_{\frac{\Delta p}{p}^{2}} \right)}{\text{dt}}|_{\text{IBS}} + \frac{d\left( \sigma_{\varphi^{2}} \right)}{\text{dt}}|_{\text{RF}} \right)$$

$$\frac{d\left( \sigma_{\frac{\Delta p}{p}^{2}} \right)}{\text{dt}}|_{\text{total}} \approx \left( 1 - \left( \frac{2\sigma_{\frac{\Delta p}{p}}}{0\text{.}\text{765}\frac{\Delta P}{P}|_{\text{sep}}} \right)^{5} \right)\left( \frac{d\left( \sigma_{\frac{\Delta p}{p}^{2}} \right)}{\text{dt}}|_{\text{IBS}} + \left( \frac{\lambda_{\text{RF}}}{2{\pi\Gamma}_{s}} \right)^{2}\frac{d\left( \sigma_{\varphi^{2}} \right)}{\text{dt}}|_{\text{RF}} \right)$$

### Longitudinal Diffusion -- continue (Luminosity lifetime backup slide)

### where

### $E = \frac{p^{2}}{2} + \Omega_{s^{2}}\left( 1 - \text{cos}\varphi \right)$ , $I = \frac{1}{2\pi}\oint_{}^{}{\text{pd}\varphi}$ and $\Gamma_{s} = \frac{\lambda_{\text{RF}}q}{2\text{πν}_{s}}\left( \alpha - \frac{1}{\gamma^{2}} \right)$ . 

**[The bunch lengthening due to RF phase noise]{.underline}**

-   At small amplitude the bunch lengthening due to RF phase noise is
    determined by its spectral density at synchrotron frequency,

$\frac{d\left( \sigma_{\varphi^{2}} \right)}{\text{dt}}|_{\text{RF}} = \pi\Omega_{s^{2}}Ρ_{\varphi}\left( \Omega_{s} \right)$
,

where the spectral density of RF phase noise is normalized as

$$\overline{\varphi_{\text{RF}^{2}}} = \int_{- \infty}^{\infty}{Ρ_{\varphi}(\omega)d\omega}$$

-   Spectral density and bunch lengthening measurement are in decent
    agreement, and they yield that

> $$P_{\varphi f}\left( \Omega_{s}/2\pi \right) = 4{\pi P}_{\varphi}\left( \Omega_{s} \right) \approx 6 \cdot \text{10}^{- \text{12}}\ \frac{\text{rad}^{2}}{\text{Hz}}$$
>
> $$\frac{d\left( \sigma_{\varphi^{2}} \right)}{\text{dt}}|_{\text{RF}} \approx \text{16}\ \frac{\text{mrad}}{\sqrt{\text{hour}}}$$

**[Intrabeam Scattering (Luminosity lifetime backup
slide)]{.underline}**

-   Smooth lattice approximation has been used for IBS to simplify the
    model

    -   Comparison with exact calculations yields coincidence within 10%

-   The following corrections has been taken into account

    -   Bunch length correction due to non-linearity of longitudinal
        focusing

    -   Average dispersion and dispersion invariant, *A~x~*, were
        calculated using lattice functions

> $\frac{d}{\text{dt}}\left( \theta_{{||}^{2}} \right) \equiv \frac{d}{\text{dt}}\left( \frac{\overline{p_{{||}^{2}}}}{p} \right) = \frac{1}{4\sqrt{2}}\frac{e^{4}N_{i}L_{C}\Xi_{||}\left( \theta_{x},\theta_{y} \right)}{m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{x}\sigma_{y}\sigma_{s}\sqrt{\theta_{x^{2}} + \theta_{y^{2}}}}$
> ,
>
> $\frac{{d\varepsilon}_{x}}{\text{dt}} = (1 - \kappa)\left\langle A_{x}\frac{{d\theta}_{{||}^{2}}}{\text{dt}} \right\rangle_{s}$
> ,
> $\frac{{d\varepsilon}_{y}}{\text{dt}} = \kappa\left\langle A_{x}\frac{{d\theta}_{{||}^{2}}}{\text{dt}} \right\rangle_{s}$
>
> where
>
> $\Xi_{||}(x,y) \approx 1 + \frac{\sqrt{2}}{\pi}\text{ln}\left( \frac{x^{2} + y^{2}}{2\text{xy}} \right) - 0\text{.}\text{055}\left( \frac{x^{2} - y^{2}}{x^{2} + y^{2}} \right)^{2}$,
>
> $\sigma_{x} = \sqrt{\varepsilon_{x}\beta_{y} + D^{2}\theta_{{||}^{2}}}$,
> $\sigma_{y} = \sqrt{\varepsilon_{y}\beta_{y}}$,
> $\theta_{x} = \sqrt{\varepsilon_{x}/\beta_{x}}$and
> $\theta_{y} = \sqrt{\varepsilon_{y}/\beta_{y}}$
>
> $$A_{x} = \left\langle \frac{D^{2} + \left( D^{'}\beta_{x} + \alpha_{x}D \right)^{2}}{\beta_{x}} \right\rangle|_{s}$$
>
> -- coupling coefficient (measurements yield that presently \~ 0.3)
> *A~x\ ~*= 19.7 cm, *~x~*=*~y~* = 48.5 m, *D* = 2.83 m

### Comparison of the Model Predictions to the Store 1953 parameters 

###  ![](media/image5.wmf)

### ![](media/image6.wmf) 

### ![](media/image7.wmf)

*Measured (solid line) and computed (dashed line) rms bunch lengths for
pbar and proton bunches*

### ![](media/image8.wmf)

*Model prediction for beam emittances and hourglass effect*

### Comments for the Store 1953

-   It is one of our best stores

-   In general there is decent agreement between predictions and
    measurements

    -   Unfortunately, our emittance measurements are not sufficiently
        accurate for detailed comparison therefore

        -   The initial values of the horizontal and vertical emittances
            were chosen to be equal

        -   The initial emittances of proton and pbar beams were
            adjusted to fit the measured luminosity and bunch lengths

-   There are two discrepancies between the model and the measurements
    at the beginning of the store

    -   The proton beam intensity is decaying faster than the model
        prediction

    -   The proton bunch length is growing slower than the model
        prediction

    ```{=html}
    <!-- -->
    ```
    -   As we figured out later it was related to incorrect values of
        the proton tunes

        -   The problem disappeared after minor tune correction in
            December 2002

    -   The most probable reason is that incorrect tunes affected the
        motion stability for particles with large synchrotron amplitudes
        which caused both

        -   Particle loss

        -   And bunch shortening (actually it was compensated by bunch
            lengthening due to IBS)

        ```{=html}
        <!-- -->
        ```
        -   Additional studies are required to figure out the actual
            reason

-   **[Conclusion]{.underline}**: for correctly tuned collider at
    present beam intensities the beam-beam effects and machine
    nonlinearity, as well as, coherent effects do not produce harmful
    effects on the beam dynamics and collider luminosity while beams are
    in collisions.

### Basic Luminosity Scenario 

Luminosity integral is calculated presuming that:

-   Machine works 46 weeks per year (6 weeks downtime or shutdown time)

-   There are 48 hour downtime per week

-   Shot setup time is 2 hour. It is not included into the downtime.

Balanced approach for both Tevatron and Antiproton source parameters

![](media/image9.wmf)

![](media/image10.wmf)

![](media/image11.wmf)

$$L = \frac{N_{p}N_{\overline{p}}f_{0}n_{b}}{2\text{πβ}^{}\sqrt{\left( \varepsilon_{\text{xp}} + \varepsilon_{x\overline{p}} \right)\left( \varepsilon_{\text{yp}} + \varepsilon_{y\overline{p}} \right)}}H\left( \frac{\sqrt{\sigma_{\text{sp}^{2}} + \sigma_{{s\overline{p}}^{2}}}}{\beta^{}} \right)$$

Efficiency of the Antiproton Recycling

  ------------------------------------------------------------- -----------
  Transverse efficiency into 30 mm mrad acceptance,*~tr~*       0.969

  Longitudinal efficiency into 3 eV s acceptance, *~L~*         0.727

  Fraction of stores with successfully decelerated protons,     0.7
  *~success~*                                                   

  Fraction of antiprotons survived at the store end, *~Tev~*    0.747

  Total efficiency of pbar recycling, *~TevsuccessLtr~*         **0.368**
  ------------------------------------------------------------- -----------

**Pbar production (1 mA10^10^** pbars)

  ------------------------------------------------------- ---------------
  Average antiproton production rate                      40 mA/hour

  Store time                                              9 hour

  Total number of stacked pbars                           360 mA

  Total number of recycled pbars                          178 mA

  Total number of pbars extracted from recycler           538 mA

  Pbar utilization factor                                 0.9

  Total number of pbars at flat top                       **484 mA**
  ------------------------------------------------------- ---------------

###  Luminosity Dependence on Antiproton Production

![](media/image13.wmf)

![](media/image15.wmf)

> *Solid lines -- no recycling*
>
> *Dashed lines -- with recycling*
>
> *Solid black line* N~pbar~=1.3510^11^ /bunch

### Effect of Luminosity Leveling and Pbar Production on the Luminosity

+---------------------------+-------+-------+------+---------+--------+
| ###                       | ###   | ###   | ##   | ### Pb  | #      |
|                           | *L~pe |  $$\i | # St | ar prod | ## $$\ |
|                           | ak~*, | nt_{} | ore  | uction, | frac{\ |
|                           |       | ^{}\t | leng |         | int_{} |
|                           | 1     | ext{L | th,  | mA/hour | ^{}\te |
|                           | 0^32^ | dt}$$ |      |         | xt{Ldt |
|                           | c     |       | ###  |         | }}{\in |
|                           | m^-2^ | fbarn | hour |         | t_{}^{ |
|                           | s^-1^ | /year |      |         | }\text |
|                           |       |       |      |         | {Ldt}| |
|                           |       |       |      |         | _{\tex |
|                           |       |       |      |         | t{basi |
|                           |       |       |      |         | c}}}$$ |
+---------------------------+-------+-------+------+---------+--------+
| ### Basic                 | ###   | ###   | ###  | ###     | ##     |
|                           |  3.31 | 3.182 | 9.00 | 40+recy | # 100% |
|                           |       |       |      | cle^\*^ |        |
+---------------------------+-------+-------+------+---------+--------+
| ### Basic                 | ###   | ###   | ###  | ###     | ###    |
| with luminosity leveling  | 2.025 | 2.796 | 8.83 | 40+recy |  87.8% |
|                           |       |       |      | cle^\*^ |        |
+---------------------------+-------+-------+------+---------+--------+
| ### Basic wit             | ###   | ###   | #    | ###     | ##     |
| h reduced pbar production |  3.31 | 2.884 | ## 1 | 30+recy | # 90.6 |
|                           |       |       | 3.10 | cle^\*^ |        |
+---------------------------+-------+-------+------+---------+--------+
| ##                        | ###   | ###   | #    | ### 40  | 89.5   |
| # Basic without recycling |  3.31 |  2.85 | ## 1 |         |        |
|                           |       |       | 3.45 |         |        |
+---------------------------+-------+-------+------+---------+--------+
| ### B                     | ###   | ###   | #    | ### 30  | 79.9   |
| asic with reduced pbar pr |  3.31 |  2.54 | ## 1 |         |        |
| oduction and no recycling |       |       | 7.95 |         |        |
+---------------------------+-------+-------+------+---------+--------+

### \* We presume that in both cases with and without recycling the shot setup time is 2 hour

![](media/image16.wmf)

### *Dependence of the luminosity and beta-function at IP for luminosity leveling scenario* 

##  2. Beam Instabilities

-   Transverse Instabilities

    -   Single bunch head-tail instability

    -   Brings many troubles into Tevatron operations

    -   Needs to be suppressed if number of protons will grow

-   Longitudinal instabilities

    -   Undamped dancing of bunches

        -   Single bunch effect

    -   Presently, it does not bring severe problems but limits desired
        shortening of the bunches

-   Significant progress was achieved in understanding of both
    instabilities during last year

-   It is not expected that multibunch instabilities will be degrading
    machine performance at the planed Run II intensities.

### Transverse Head-Tail Instability

-   In the most detailed recent measurements ( P. Ivanov, V. Scarpine),
    all the features of the head-tail instability were observed

    -   The instability threshold does not depend on the number of
        bunches;

    -   When the instability is developed, the transverse amplitudes and
        phases have correct modulation along the bunch

-   At injection, the growth rates **\~150 s^-1^** for *N*=2.510^11^ per
    bunch, compared to **500 s^-1^** of the synchrotron angular
    frequency.

-   To suppress CI, high chromaticities are required, which deteriorate
    the beam lifetime

![](media/image17.wmf)

**[Head-tail Instability Simulations]{.underline}**

-   A Monte-Carlo C++ code has been written (A. Burov) to find the
    impedance responsible for the instability. It allows to use

    -   any given wake function(s),

    -   arbitrary longitudinal distribution

    -   and initial conditions,

    -   and to see the transverse beam dynamics in time domain.

-   The code was tested for the air-bag longitudinal distribution, where
    the analytical results are known, and the perfect agreement has been
    found.

-   Performing simulations

    -   with the Gaussian longitudinal distribution (close to real one,
        *~s~* = 1 m)

    -   and with the impedance of resistive wall type,
        $Z_{}(\omega) \propto \frac{(1 + i)}{\sqrt{\omega}}$

> we obtained that the resistive part of total impedance is about or
> above 4 M/m at the bunch frequency of 50 MHz

-   Presently, the model does not take into account the Landau damping,
    and therefore we can only quote the low boundary of the impedance

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```
-   This is

    -   4 times higher than the impedance of vacuum chamber resistive
        walls,

    -   and 2.3 times above Run IIA handbook estimate.

**Where Is the Lost Transverse Impedance?**

-   Transverse impedance of the Lambertson magnets compared to the
    chamber resistive one is

$$\frac{Z_{L}}{Z_{c}} \approx \frac{l_{L}\sqrt{\mu}}{C}\sqrt{\frac{\rho_{L}}{\rho_{c}}}\frac{b_{L}}{d}\frac{b_{c^{3}}}{b_{L^{3}}}$$

> *b~L~*, *b~c~* are the distances to the resistive surface,
>
> *~L~*, *~c~* are the resistivities,
>
> is the magnetic permeability,
>
> *l~L~* is the occupied Lambertson magnet length,
>
> *C* is the circumference,
>
> *d* is the lamination periodicity,
>
> and the superscripts L, c relates to the Lambertson magnets and the
> main chamber

-   Substituting

*~c~* / *~L~* = 4, *b~L~* = 10 mm, *b~c~* = 25 mm, d = 1 mm, *l~L~* =
82.8 = 22.4 m, and = 100

leads to *Z~L~* / *Z~c~* 2.8 , or ***Z~L~* 2.8 M/m**.

**What Can Be Done?**

-   C0 Lambertson magnet has been removed from the Tevatron at this
    (January) shutdown.

    -   This has to give 30-50% of the impedance reduction.

-   Inner surfaces of the remaining F0 Lambertson magnets are planned to
    be shielded from the beam in the summer

    -   That should further reduce the Tevatron transverse impedance to
        about 1 M/m and should greatly reduce problems with the
        head-tail instability

-   There are two additional leverages which we already exercised and
    which can be used if necessary

    -   Transverse bunch-by-bunch damper

        -   While it is designed to damp only dipole mode it also damps
            the head-tail instability because in the case of non-zero
            chromaticity head-tail modes also have non-zero dipole
            moment

    -   Introducing cubic (octupole) non-linearity suppresses the
        instability due to increasing the betatron tune spread

        -   The drawback of this solution is amplified sensitivity of
            tunes to the closed orbit changes

### Longitudinal Single Bunch Instability

### ('Dancing bunch' effect)

-   Long-term coherent synchrotron oscillations of proton bunches are
    observed in Tevatron.

-   Bunch shape at the oscillations differs for uncoalesced and
    coalesced bunches.

    -   Uncoalesced bunches - oscillations persist for hours

    -   Coalesced bunches -- oscillations are damped during about 5
        minutes

        -   Longitudinal bunch-by-bunch damper accelerates the damping

![](media/image18.wmf)

-   The synchrotron tune spread would damp oscillations within seconds
    without coherent interactions inside bunches

**The following below data relate to the uncoalesced bunches at 150
GeV.**

**Bunch waveforms at 150 GeV, 210^11^ protons in 30 bunches**

![](media/image19.png){width="4.2444444444444445in"
height="2.479861111111111in"}

*40 turns from injection to 1-st scan and between scans*

![](media/image20.png){width="4.2444444444444445in"
height="2.479861111111111in"}

*\~1.5 min between colors. Each color includes 8 scans in 40 turns. Hor.
axis - RF phase/2.*

*Integer - bunch number in the batch.*

-   Effect of **inductive** longitudinal impedance separates coherent
    and incoherent tunes and prevents decoherence at (V.Balbekov,
    S.Ivanov, 1991)

> \|\| \> ~c~
>
> where is synchrotron tune spread,
>
> ~c~ - coherent tune shift produced by the impedance.

-   For Tevatron at 150 GeV it yields:

> \|Z/n\|\[\] \> 2·10^115^/*N* 1

*N* = 10^10^ - number protons per bunch,

= 0.5 - bunch half-length in RF radian.

-   Run IIA handbook estimate yields very close result at characteristic
    n \~ 1000:

> $$\frac{Z}{n}\lbrack\Omega\rbrack \approx i + (1 + i)\frac{\text{20}}{\sqrt{n}}$$

-   But due to very strong dependence of the instability threshold on
    bunch length the coincidence should be considered rather qualitative
    than quantitative

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-   While removing and shielding Lambertson magnets significantly
    changes transverse impedance an effect on longitudinal impedance is
    much smaller and we do not expect significant changes of the
    longitudinal bunch dynamics after Lambertson removal

-   It is expected that the longitudinal instability will prevent the
    bunch shorter then \~ 50 cm!!!

    -   More studies are required

### 3. Beam-Beam effects

-   Two types of the beam-beam
    effects![](media/image22.wmf){width="3.9118055555555555in"
    height="3.9118055555555555in"}

    -   Head-on

        -   Run IB proton bunch population of \~2.710^11^ proton/bunch
            was set by the head-on collisions

        -   We aim to achieve the same number of protons per bunch

            -   Linear beam-beam tune shift 0.02 for two interaction
                points

            -   Tunes are between 5-th, 7-th and 12-th order resonances

    -   Long range

        -   Much stronger than for Run IB

        -   Additional tune spread within one bunch

            -   510^-3^

        -   Tune spread between bunches (*N~p~=2.710^11^*)

            -   At injection: ~x~ 510^-3^, ~y~ 2.510^-3^

            -   At flat top: ~x~ ~y~ 810^-3^

        -   Presently, the long range beam-beam effect is the major
            limitation of proton intensity increase

-   **Long range beam-beam effects**

    -   For fixed separation in 's the tune shift does not depend on
        energy![](media/image23.wmf){width="4.384027777777778in"
        height="2.848611111111111in"}![](media/image24.wmf){width="4.386111111111111in"
        height="2.879166666666667in"}

        -   It requires U Energy

    -   High voltage separators are maxed-out at \~ 500 GeV

        -   That reduces the beam separation at the end of acceleration
            by 1.4 times

    -   Acceleration and squeeze are the most sensitive steps from the
        beam-beam effects point of view

-   What can be done to alleviate harmful effects of long-range
    collisions

    -   Better optimization of helical orbits

    -   Optics change in A0 should additionally help with beam
        separation

    -   Increasing voltage on the separators

> $$\Delta\nu \propto \frac{1}{a^{2}}$$

-   Adding new separators is also possible

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```
-   Reducing initial emittance of the pbar beam

-   Tevatron electron lens can reduce both long range and head-on tune
    shifts

**Collective stability and long-range beam-beam effects**

-   Long-range beam-beam effects modify machine chromaticity so that
    different bunches see different chromaticities

![](media/image25.png)

-   If not addressed the spread of chromaticities (\~ 10 units for final
    Run II parameters) can cause

    -   The head-tail transverse instability for one or few bunches

    -   It affects the instability suppression by the transverse damper
        due to the fact that at certain chromaticities an internal
        motion in the bunch becomes uncoupled with motion of the bunch
        center of gravity

**[132 ns scenario]{.underline}**

-   There is considerable risk and uncertainty even in increasing the
    proton bunch population to 2.710^11^ /bunch for present 36 by 36
    bunch operating scenario

-   132 ns scenario implies

    1.  tripling the number of bunches

        and tripling the total proton charge in the ring

    -   It will increase the strength of long-range beam-beam effects by
        additional factor of three so that it will be 5 times higher
        than presently achieved total proton current

        -   We do not know how we could deal with long range beam-beam
            effects at such proton beam intensity

-   Taking into account that 3 fbarn/year can be achieved with 36x36;
    and the luminosity leveling can be used to limit the peak luminosity
    we believe that we need to stop further work on the 132 ns scenario

### 4. Particle Loss at Injection

## Experimental observations

-   Proton lifetime at proton helix (1-- 4 hour) is much worse than at
    central orbit (\~10 hour)

-   Lifetime is affected by the machine chromaticity

    -   Smaller chromaticity improves the lifetime but its reduction is
        limited by head-tail instability

        -   Using octupoles to stabilize the instability improves the
            lifetime but makes machine tuning oversensitive to the orbit
            changes due to the tune dependence on orbit

-   Strong dependence of the lifetime on bunch length

-   Intensity lifetime is much worse than the emittance lifetimes

    -   Proton intensity decays as $N\left( 1 - \sqrt{t/\tau} \right)$

-   Additionally to mentioned above, the **pbar lifetime** is strongly
    affected by beam-beam effects

## Basic mechanisms and reasons of the proton loss

![](media/image27.wmf)

## Effects of longitudinal diffusion due to IBS and RF noise are amplified by 

## Overfilled bucket at injection

-   Shallowing the potential well near separatrix

-   Instability of particle motion at large synchrotron amplitudes

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```
-   Effects of transverse diffusion create particle loss due to

    -   aperture limitations

    -   and reduced dynamic aperture for particles with large
        synchrotron apertures

-   Major transverse diffusion mechanisms are

    -   the residual gas scattering
        (${d\varepsilon}_{x}/\text{dt}|_{\text{Gas}} \approx {d\varepsilon}_{y}/\text{dt}|_{\text{Gas}} \approx 1\text{.}1\ \text{mm}\ \text{mrad/hour}$)

    -   and IBS
        (${d\varepsilon}_{x}/\text{dt}|_{\text{IBS}} + {d\varepsilon}_{y}/\text{dt}|_{\text{IBS}} \approx 1\text{.}2\ \text{mm}\ \text{mrad/hour}$)

**Measured dependence of proton beam intensity on time at injection**

![](media/image31.png)

![](media/image32.wmf)

*Measured and used in simulations initial longitudinal particle density*

**[Numerical simulations of proton/pbar dynamic apertures for different
chromaticities]{.underline}**

-   The above simulations take only longitudinal particle loss from the
    bucket

-   They produce close results but do not address the lifetime
    dependence on chromaticity

    -   Most probable reason

        -   Reduction of dynamic aperture for particles with large
            amplitudes due to an increase machine non-linearity with
            chromaticity

![](media/image34.png)

*All known non-linearities are included for protons (~p~ = 25 mm mrad)*

-   The simulations do not exhibit significant effect on the dynamic
    aperture while measurements point out that it should be there

    -   We ignore orbit displacement in simulations

    -   We are not sure that all the non-linearities are correctly
        included

    -   More studies both theoretical and experimental are required

### Both transverse and longitudinal losses cause $\text{exp}\left( - \sqrt{\frac{t}{\tau}} \right)$ intensity decay if the beam distribution is close to the aperture limit

### We do not know a ratio of transverse to longitudinal losses but it is more probable that the longitudinal losses dominated the proton loss at injection 

### After removing C0 lambertson we are gaining the transverse aperture and the longitudinal losses will almost certainly dominate for protons

### Therefore having sufficiently small longitudinal emmitance of proton beam coming from MI is the only way to prevent proton losses at injection and ramp

### Taking into account the shallowing of the potential well we can write a simple estimate for a fraction of longitudinal emittance need to be free of particles

> $$\frac{\Delta\varepsilon}{\varepsilon_{\text{max}}} \approx 2\sqrt{\frac{T}{\varepsilon_{\text{max}}}\frac{d\varepsilon}{\text{dt}}}$$
>
> IBS determines that
> $\varepsilon_{\text{max}}/\left( d\varepsilon/\text{dt} \right)$ 50
> hour
>
> Then for T = 0.5 hour we obtain
> $\Delta\varepsilon/\varepsilon_{\text{max}}$ 0.2 which means that
> \~100% of particles have to be within 90% of bucket size or ~s~ 3.8 eV
> s

### 5. The Source of Background in the Particle Physics Detectors

-   Two sources of the background

    -   Single scattering

        -   Gas scattering

        -   Touschek effect (IBS)

    -   Multiple scattering and diffusion

        -   Gas scattering

        -   IBS

        -   Diffusion due to beam-beam effects

        -   RF noise

-   Most of the particles (\~90%) are lost due to multiple scattering
    and diffusion

    -   That determines that the protection from multiple scattering is
        more important

-   Tevatron collimation system was optimized to intercept particles
    with slowly growing amplitude

    -   It is based on a set of two primary and four secondary
        collimators to intercept background from both proton and pbar
        beams

        -   Each collimator is designed to scrape particles vertically
            and horizontally from one side

    -   The system is very effective to intercept particles with slowly
        growing amplitude,   210^-4^

        -   But it does not do a good job to intercept single scattered
            particles, 0.02

> ![](media/image36.png)
>
> *Simulated dependence of beam loss in CDF and D0 detectors as function
> of average Tevatron vacuum*

-   Recently performed simulations verified that the single scattered
    particles create major fraction of the detector background

    -   Tevatron vacuum has been improved after summer shutdown and it
        improved background in detectors

-   Planned luminosity growth by an order of magnitude will also cause
    growth of the background but in significantly smaller scale

    -   The background will be determined by vacuum and if vacuum will
        stay the same the background will grow by 2.8 times

**Reduction of the background**

-   We plan further vacuum improvements

-   A system of shadow collimators placed at A48 and C48 straight
    sections in front of the CDF and D0 (upstream of the last three
    dipoles before the IP) allows to suppress this background by an
    order of magnitude

> ![](media/image37.png)
>
> **[Conclusions]{.underline}**

1.  During last year we made significant progress in understanding of

    -   Beam heating mechanisms and luminosity lifetime

    -   Transverse and longitudinal instabilities

    -   Sources of background in particle physics detectors

2.  We are sill far away to understand all details of the beam-beam
    effects and it brings most of uncertainty in predictions of the
    final Run II luminosity

3.  To make progress in understanding of the beam-beam effects we will
    need significant contribution into

    -   instrumentation,

    -   theory

    -   and experimental studies.