# Presentation: 616157

## Rescattering Effects on HBT Interferometry

- <date/time>

- Joseph Kapusta Yang Li (grad student)
- University of Minnesota

## What does HBT measure?

- <date/time>

- For stars it measures the radius.  Rescattering in the photosphere is irrelevant since the thickness is so small compared to the radius.  Opaque.
- For elementary particle collisions it measures the space-time distribution of the production points.  Rescattering effects are usually negligible.  Transparent.

## What does HBT measure?

- <date/time>

- For heavy ion collisions, rescattering after production may be significant.  The source is neither opaque nor transparent.  The usual interpretation is that HBT measures the distribution of last scatterings.  What if the last scattering is very soft?  What counts as the “last scattering”?

## Assumptions:

- <date/time>

- Nonrelativistic
- Localized wave-packet produced at tP and momentum eigenstate observed at t2>> tP
- Different wavepackets are produced with uncorrelated phases
- No interactions between produced particles, only with static localized potentials

##   produced(x) = d3p/(2)3 (p,xP) exp{i(xP)} exp{i[p•(x-xP) - Ep(t-tP)]}

- Wave packet with phase  centered at **x**P at time tP.
-  Momentum space wave function  may be chosen real.

-           A = d3x1d3x2 *obs(**x**2) G(**x**2,t2; **x**1,t1) prod(**x**1)

- observed (**x**) = exp{i[**k**•**x** - Ekt2)]}

- Momentum eigenstate observed by detector at time t2 >> tP.

- Single particle amplitude with G = propagator
-  that includes the effects of rescattering.

## C(k1,k2) = 1 + |d4xP exp(iqxP) eff(k1,k2;xP)|2

- C(**k**1,**k**2) = 1 + |d4xP exp(iqxP) eff(**k**1,**k**2;xP)|2

- eff = [K(**k**1;xP;t2,t1)P(**k**1)] [K*(**k**2;xP;t2,t1) P(**k**2)] (xP)

- K(**k**;xP;t2,t1) =  d3p/(2)3 exp[ixP (p-k1)] exp[-iEpt1]
-                          * G(**k**,t2;**p**,t1) (**p**,xP)

- P(**k**) =  d4xP (xP) |K(**k**;xP;t2,t1)|2

- The single particle momentum distribution.

- The effective density of wavepacket sources.

## When there is no rescattering

- <date/time>

-  K = Kfree and eff = .
- Standard results are reproduced.

- As a first illustration consider the
-  one dimensional problem.

## produced(x;t1)

- produced(x;t1)

- observed(x;t2) = exp[i(kx-Ekt2)]

- k

- k

- -k

- k

- k

## Slide 9

## When the transmission coefficient is much greater than

- the reflection coefficient backscattering can be neglected.
- K/Kfree = 1 - imvFT(0)/2k - m2v2FT(0)/2k2

- Since this is independent of xP the correlation function
- is unaffected by rescattering and eff = .  This is true
- even when backscattering is included.

- In one dimension the HBT correlation function measures
- the size of the production source, where the wavepackets
- are produced, and is not affected by elastic rescattering
- on static localized potentials.

## scat(xc) = Nc (2 Rc)3 exp[-xc2/2Rc2]

-  of wavepacket source

- scat(**x**c)** = **Nc (2 Rc)3 exp[-**x**c2/2Rc2]

- and scattering potentials

- (x)** = **1(42RP3t) exp[-**x**2/2RP2] exp[-t2/2t2]

- with the Fourier transform of the potential parametrized as

- vFT(**q**) = vFT(0) exp[-**q**2r02]

## Rout2 = RP2{1+

-  size via the curvature in momentum space we find

- d(0)
-  d

- Rout2 = RP2{1+

- 2 k4

-           
- (1+2)(1+)2

- scat2(0)

- }

- k2
- m2

- +

- t2

- Rside2 = RP2{1

- 4
- k2

-    RP2
- (1+2)2

- scat2(0)

- d(0)
-  d

- }

-  = RP2  Rc2

- valid at high momentum and to first order in the cross section.

## Conclusions:

- <date/time>

- We calculated elastic rescattering effects on the HBT correlation function.
- Of crucial importance is the space-time position of the wavepacket when its phase is taken to be random.
- In a model calculation Rout increases and Rside decreases but its generality is not yet known.
- Obvious generalizations to moving potentials, resonances, relativistic motion, fermions, partial coherence, flow, and so on.